3-torsion Subgroups of Jacobians of Plane Quartics (2024)

Elvira LupoianandJames Rawson

Abstract.

In this paper we give an algorithm to find the 3-torsion subgroup of the Jacobian of a smooth plane quartic curve. We describe $3-$ torsion points in terms of cubics which triply intersect the curve, and use this to define a system of equations whose solution set corresponds to the coefficients of these cubics.We compute the points of this zero-dimensional, degree $728$ scheme first by approximation, using hom*otopy continuation and Newton-Raphson, and then using lattice reduction or continued fractions to obtain accurate expressions for these points.We describe how the Galois structure of the field of definition of the $3$ -torsion subgroup can be used to compute local conductor exponents, including at $p=2$ .

Key words and phrases:

Torsion, Jacobians, Plane Quartics, Conductors, LLL, hom*otopy Continuation

2000 Mathematics Subject Classification:

11Y60 (primary), 11G30, 11Y50

Both authors are supported by the Warwick Mathematics Institute Center for Doctoral Training, and gratefully acknowledge the funding from the UK Engineering and Physical Sciences Research Council. The first named author is supported by the grant EP/V520226/1. The second named author is supported by the grant EP/W523793/1.

1. Introduction

The celebrated result of Mazur [17] gives a complete classification of rational torsion subgroups of elliptic curves. The general study of torsion of abelian varieties is considered out of reach. An interesting variant of this question could be the study of $n-$ torsion points, for a fixed natural number $n$ .For Jacobians of genus 1 curves, this is a largely elementary problem, since explicit equations are known for the Jacobian (as it is an elliptic curve) and the group law is accessible. For Jacobians of higher genus curves, the problem becomes much more difficult. Perhaps the clearest description is that of $2-$ torsion, since for the hyperelliptic curves, it can be easily deduced from the Weierstrass model, and for non-hyperelliptic curves $2$ -torsion points correspond to multi-tangent hyperplane to the curve, as described in [11]. In [5], an explicit description of $3-$ torsion is derived for Jacobians of genus $2$ curves. This is generalised in [15], for hyperelliptic curves of genus $3$ . For hyperelliptic curves, such description of torsion can be obtained using a Weierstrass model for the curve and careful analysis of Riemann-Roch spaces. However, when the curve is not hyperelliptic, the description of torsion points tends to become more geometric.

Explicitly computing such torsion subgroups has important applications. For instance, it’s used for descent purposes in [5] and [6]; and also for conductor computations in [10].

In this note, we give a description of the 3-torsion of genus 3, non-hyperelliptic curves in terms of degree 3 curves meeting the original quartic in 4 places, each with multiplicity 3. As a consequence of this description, we are able to give equations for all of the $728=3^{6}-1$ unramified covers of the curve with Galois group $A_{3}$ .

After rigidifying the configuration of a cubic and the genus 3 curve, we give a schema for calculating all such curves. Using numerical methods (hom*otopy continuation and Newton-Raphson), we find high-precision complex approximations to the cubics. The coefficients can be recognised as algebraic integers by algorithms like Lenstra-Lenstra-Lovász (LLL) or continued fraction methods on the minimal polynomial. The 3-torsion subgroup computed is then verified by reduction modulo primes. A similar method of computing torsion is described in [16] and [15].

We conclude by providing detailed examples in the cases of the Fermat quartic, $x^{4}+y^{4}=1$ , and the Klein quartic, $x^{3}y+y^{3}+x=0$ . For these curves we compute the conductor exponents by torsion methods, and compare against the value computed from their structure as modular curves. Symbolic calculations were carried out using the Magma software package [3] and numerics with Julia [1].

Our source code is available from the GitHub repository:

https://github.com/ElviraLupoian/PlaneQuartics3Torsion

2. Derivation

From now on, $C$ will be a genus $3$ , non-hyperelliptic, smooth, projective, geometrically integral curve, embedded into $\mathbb{P}^{2}$ via the canonical embedding. Denote by $J$ the Jacobian variety attached to $C$ . We note that $J$ is isomorphic to the Picard group, and we regard its points as equivalence classes of degree zero divisor on the curve. Let $[D]$ be a 3-torsion point of the Jacobian. As $D$ is degree 0, $K_{C}+D>0$ , so after fixing a choice of canonical divisor, $K$ , $D$ can be written as $D^{\prime}-K$ , where both $D^{\prime}$ and $K$ are effective. By assumption, $3D\sim 0$ , so $3D^{\prime}\sim 3K_{C}$ . Sections of $3K_{C}$ are cubics, and so $3D^{\prime}$ is the intersection of $C$ with a plane cubic. As $3D^{\prime}$ is the triple of an effective divisor, the cubic must meet $C$ with multiplicity (a multiple of) 3 at every intersection point.

The divisor $D^{\prime}$ has degree 4, and so moves ( $l(D^{\prime})\geq 2$ ). We may, therefore, assume $D^{\prime}$ contains a flex. By change of coordinates, it can further be assumed this flex is at $(X:Y:Z)=(0:1:0)=P$ with tangent line $Z=0$ . Functions vanishing to order 3 at this point lie in the ideal $(Z,X^{3})$ , and so the class of cubics to be considered is those in long Weierstrass form (in the affine chart $Z=1$ ).

Not all cubics of this form give rise to non-trivial torsion elements, however. If $D^{\prime}-K\sim 0$ , then $D^{\prime}$ is a canonical divisor and so the intersection of $C$ with a line, and it contains $P$ . The only lines through this point are the lines of constant $x$ , hence cubics that are $(x-a)^{3}$ must be ruled out. One way to do this is to assume that at least one coefficient of $y$ is non-zero, although this may exclude some desired cubics.

If $D^{\prime}-P$ is not a special divisor, then this representation is unique; if $D_{1}^{\prime}-K\sim D_{2}^{\prime}-K$ , then $D_{1}^{\prime}-P\sim D_{2}^{\prime}-P$ , and so there is equality of divisors as neither moves. As the divisors are the same, the cubics cutting them out must be the same up to scaling.

If $D^{\prime}-P$ is a special divisor, then it can be written as $K^{\prime}-Q$ , for some canonical divisor $K^{\prime}$ and point $Q$ . It follows that, up to linear equivalence, $D=P-Q$ , and so $3D=3P-3Q\sim 0$ . This shows $Q$ is a flex. The tangent to $Q$ meets $C$ at another point, $R$ , and the divisors equivalent to $3Q$ are the intersection of $C$ with lines through $R$ . As $3P\sim 3Q$ , the tangent to $P$ must meet $C$ again at $R$ . So as long as $C$ has a flex whose corresponding flex line does not meet another in this way, this method will compute all non-trivial three torsion, and find unique expressions for each element.

This method can be adapted for $P$ not a flex, for instance if such a flex does not exist or to make use of a rational point. In this case, the cubics to be considered are those of the form $a_{1}(y^{2}-\alpha x^{2}y)+a_{2}xy+a_{3}y+a_{4}x^{3}+a_{5}x^{2}+a_{6}x+a_{7}$ , where $\alpha$ is a constant depending on the curvature of $C$ at $P$ .Thus we have proved the following.

Theorem 1.

Let $C$ be a smooth, projective, geometrically integral, non-hyperelliptic curve of genus $3$ , defined over a number field and canonically embedded in $\mathbb{P}^{2}$ . Assume that $C$ has a flex at $(0:1:0)$ , with tangent line $Z=0$ , then the $3-$ torsion points of it’s Jacobian correspond to cubics of the form

$a_{1}y^{2}+a_{2}xy+a_{3}y+a_{4}x^{3}+a_{5}x^{2}+a_{6}x+a_{7}$

which intersect the curve with multiplicities divisible by $3$ . Moreover, this correspondence preserves the Galois action.

This description of $3$ -torsion points allows us to derive equations, whose solution set represents coefficients of cubics of the above form.

2.1. Scheme of $3$ -torsion points

Let $C$ be a curve as above, with an affine chart given by $f\left(x,y\right)=0$ . Suppose that cubics corresponding to $3$ -torsion points on the Jacobian $J$ are of the form

$\alpha_{1}y^{2}+\alpha_{2}xy+\alpha_{3}y+\alpha_{4}x^{3}+\alpha_{5}x^{2}+%\alpha_{6}x+\alpha_{7}$

and we assume that at least one of $\alpha_{1},\alpha_{2},\alpha_{3}$ is non-zero. We describe the case when $\alpha_{1}\neq 0$ and derive a system of equations whose solutions parameterise $\alpha_{2},\ldots,\alpha_{7}$ . Other cases are very similar.

We want to find $\left(a_{1},\ldots,a_{6}\right)$ such that the plane cubic defined by

$y^{2}+a_{1}xy+a_{2}y+a_{3}x^{3}+a_{4}x^{2}+a_{5}x+a_{6}$

intersects the curve at $3$ points, each with multiplicity $3$ . Observe that the above gives an expression $y^{2}=g\left(x,y\right)$ , and the monomials of $g$ are not divisible by $y^{2}$ . The equation defining our affine quartic can be viewed as an equation in $y$ , and it’s of the from

$f(y)=s_{1}(x)y^{4}+s_{2}(x)y^{3}+s_{3}(x)y^{2}+s_{4}(x)y+s_{5}(x)$

with $s_{i}(x)\in\mathbb{Q}[x]$ . The intersection of the two is be described by

	$\displaystyle f(y)$	$\displaystyle=s_{1}(x)g(x,y)^{2}+s_{2}(x)g(x,y)y+s_{3}(x)g(x,y)^{2}+s_{4}(x)y+%s_{5}(x)$
		$\displaystyle=t_{1}(x)y^{2}+t_{2}(x)y+t_{3}(x)$

for some $t_{i}\in\mathbb{Q}[a_{1},\ldots,a_{6}][x]$ . We repeat this to eliminate the $y^{2}$ monomial in the above expression to obtain

$f(y)=u_{1}(x)y+u_{2}(x)$

and hence the above gives us an expression for $y$ . Substituting $-u_{2}/u_{1}$ for $y$ in the cubic gives us a degree $9$ polynomial $h(x)$ , and to ensure that the points of the intersection of our cubic and quartic occur with multiplicity $3$ , we require $h$ to be the cube of a degree $3$ polynomial in $x$ , that is,

$h(x)=a_{10}(x^{3}+a_{7}x^{2}+a_{8}x+a_{9})^{3}$

and equating the coefficients of the above expression gives $10$ equations in $a_{1},\ldots,a_{10}$ . Moreover, to avoid singular solutions, we require that $u_{1}$ and $u_{2}$ have no common factors, which can be imposed by adding the equation

$a_{11}\text{Res}\left(u_{1},u_{2}\right)+1=0$

to our system, where $\text{Res}(u_{1},u_{2})$ denotes the resultant of the two polynomials.

Theorem 2.

Let $C$ be a curve as in theorem 1. The subgroup of $3-$ torsion points of its Jacobian is explicitly computable by a finite algorithm.

3. Approximations and hom*otopy Continuation

Let $f_{1},\ldots f_{n}$ be equations defining a zero-dimensional scheme, the points of which give the coefficients of our cubics. In this section we give an overview of how to obtain high precision approximations using the classical Newton-Raphson method and hom*otopy continuation. We begin with a brief summary of the Newton-Raphson method, a comprehensive overview can be found in [21, Page 298].

Consider the $\mathbb{C}^{n}$ valued function defined by $f_{1},\ldots,f_{n}$ ,

$F=\left(f_{1},\ldots,f_{n}\right):\mathbb{C}^{n}\longrightarrow\mathbb{C}^{n}$ .

Let $dF$ be the Jacobian matrix of $F$ and suppose $\mathbf{x}_{0}$ is an approximate solution of $F$ with $dF\left(\mathbf{x}_{0}\right)$ invertible.

For each integer $k\geq 1$ , define

$\mathbf{x}_{k}=\mathbf{x}_{k-1}-dF\left(\mathbf{x}_{k-1}\right)^{-1}F\left(%\mathbf{x}_{k-1}\right)$ .

If the initial approximation $\mathbf{x}_{0}$ is good enough, the resulting sequence $\{\mathbf{x}_{k}\}_{k\geq 0}$ converges to a root of $F$ , with each iterate having increased precision. For our computations, we efficiently implemented Newton-Raphson in Julia.

We compute the required initial approximations using hom*otopy continuation and its implementation in Julia (see [4]).

3.1. hom*otopy Continuation

hom*otopy continuation is a method for numerically approximating the solutions of a system of polynomial equations by deforming the known solutions of a similar system. In this subsection we give a sketch of the idea, a detailed explanation of this theory can be found in [23] .

We denote by $\text{deg}\left(F\right)=\prod_{i}\text{deg}\left(f_{i}\right)$ , where $\text{deg}\left(f_{i}\right)$ is the maximum degree of the monomials of $f_{i}$ . Let $G$ be a system of $n$ polynomials in $a_{1},\ldots,a_{n}$ , with exactly $\text{deg}\left(F\right)$ solutions, all of which are known or easily computable. This will be known as a start system. The standard hom*otopy of $G$ and $F$ is a function

	$\displaystyle H:\mathbb{C}^{n}\times\left[0,1\right]\longrightarrow\mathbb{C}^%{n}$
	$\displaystyle H\left(\mathbf{x},t\right)\ =\ \left(1-t\right)G\left(\mathbf{x}%\right)+tF\left(\mathbf{x}\right)$

Fix $N\in\mathbb{N}$ , and define $H_{s}\left(\mathbf{x}\right)=H\left(\mathbf{x},s/N\right)$ for $s\in\left[0,N\right]\cap\mathbb{N}$ . For $N$ large enough, the solutions of $H_{s}\left(\mathbf{x}\right)$ are good approximations of the solutions of $H_{s+1}\left(\mathbf{x}\right)$ , and their precision can be increased by using Newton-Raphson. The solutions of $H_{0}\left(\mathbf{x}\right)=G\left(\mathbf{x}\right)$ are known, and they define paths to approximate solutions of $H_{N}\left(\mathbf{x}\right)=F\left(\mathbf{x}\right)$ .

Notably, given any system $F$ , a start system (and its solutions) can always be computed. hom*otopy continuation is efficiently implemented in the Julia package hom*otopyContinuation.jl (see [4]). This implementation gives approximates solutions which are accurate to 16 decimal places. For our computations we used the approximate solutions and 200 iterations of Newton-Raphson to obtain an accuracy of 600 decimal places.

4. Lattice-Reduction and Continued Fractions Techniques

Let $\mathbf{x}=\left(x_{1},\ldots,x_{n}\right)$ be a point on a scheme for which we have a complex approximation $\mathbf{a}=\left(a_{1},\ldots,a_{n}\right)$ of arbitrarily large precision $k$ .

One way of computing minimal polynomials is by searching for short vectors in an appropriately defined lattice. Short vectors can be computed using the Lenstra-Lenstra-Lovász lattice reduction algorithm (LLL) which, given a lattice, returns a reduced basis whose vectors have small norms, see [7, Section 2.6]. This is a standard technique for computing short vectors, for instance see [7, Section 2.7.2] or [19, Chapter 6.1]. To compute an LLL- basis we use the Julia implementation provided by LLLplus.jl.

When computing the shortest vector in an appropriately defined lattice is impractical, we can also attempt to find the minimal polynomial by using our approximations and continued fractions to compute the $\mathbb{Q}$ -rational polynomial whose roots are all of the $i$ th coordinates of points on the scheme.

4.1. Lattice Reduction and Minimal Polynomials

We give an overview of the theory required to compute our minimal polynomials. This was necessary in our computations as we made use of the Julia’s LLL implementation, which was significantly faster then using Magma’s built in commands, such as MinimalPolynomial().

Fix $1\leq i\leq n$ and let $\theta=x_{i}$ and $\alpha=a_{i}$ . We want to find $d_{\theta}\in\mathbb{N}$ and $c_{0},\ldots,c_{d_{\theta}}\in\mathbb{Z}$ such that

$c_{d_{\theta}}\theta^{d_{\theta}}+\ldots+c_{1}\theta+c_{0}=0$ .

We assume that the imaginary part of $\alpha$ is very small, and hence we are probably approximating a real number $\theta$ . At the end of this section, we explain how the method described below can be adapted when $\theta$ is not real.

Let $\alpha=\text{Real}\left(\alpha\right)\in\mathbb{R}$ and fix a constant $C=10^{k^{\prime}}$ , for some large natural number $k^{\prime}<k$ , such that

$\mid\left[C\cdot\alpha^{i}\right]-C\cdot\theta^{i}\mid\ \leq\ 1$ for all $0\leq i\leq d_{\theta}$

where $\left[x\right]$ denotes the integer part of a real number $x$ . Let $\mathcal{L}$ be the lattice generated by the columns of the $\left(d_{\theta}+1\right)\times\left(d_{\theta}+1\right)$ matrix

$A=\begin{pmatrix}1&\ldots&0&0\\0&\ldots&0&0\\\vdots&\ddots&\vdots&\vdots\\0&\ldots&1&0\\\left[C\alpha^{d_{\theta}}\right]&\ldots&\left[C\alpha\right]&\left[C\right]\\\end{pmatrix}=\left(v_{d_{\theta}},\ldots,v_{1},v_{0}\right).$

Define $\gamma=c_{d_{\theta}}\left[C\alpha^{d_{\theta}}\right]+\ldots+c_{1}\left[C%\alpha\right]+c_{0}\left[C\right]$ and observe that

$\mathbf{c}=\begin{pmatrix}c_{d_{\theta}}\\\vdots\\c_{1}\\\gamma\end{pmatrix}=c_{d_{\theta}}v_{d_{\theta}}+\ldots+c_{0}v_{0}\in\mathcal{%L}.$

The vector of coefficients of the required polynomial $\mathbf{c}_{\infty}=\left(c_{d_{\theta}},\ldots,c_{0}\right)^{T}$ can be recovered from $\mathbf{c}$ , by setting

$c_{0}=\frac{1}{C}\left(\gamma-\left(c_{d_{\theta}}\left[C\alpha^{d_{\theta}}%\right]+\ldots+c_{1}\left[C\alpha\right]\right)\right)$ .

The norm of $\mathbf{c}$ is bounded by the norm of $\mathbf{c_{\infty}}$ as follows,

	$\displaystyle\|\|\mathbf{c}\|\|$	$\displaystyle=\sqrt{c_{d_{\theta}}^{2}+\ldots+c_{1}^{2}+\gamma^{2}}$
		$\displaystyle\leq\sqrt{c_{d_{\theta}}^{2}+\ldots+c_{1}^{2}+\left(\gamma-Cc_{d_%{\theta}}\theta^{d_{\theta}}-\ldots-Cc_{1}\theta-Cc_{0}\right)^{2}}$
		$\displaystyle\leq\sqrt{c_{d_{\theta}}^{2}+\ldots+c_{1}^{2}+\left(c_{d_{\theta}%}(\left[Ca^{d_{\theta}}\right]-C\theta^{d_{\theta}})+\ldots+c_{1}(\left[Ca%\right]-C\theta)+c_{0}(\left[C\right]-C)\right)^{2}}$
		$\displaystyle\leq\sqrt{c_{d_{\theta}}^{2}+\ldots+c_{1}^{2}+\left(c_{d_{\theta}%}+\ldots+c_{1}+c_{0}\right)^{2}}$
		$\displaystyle\leq\sqrt{c_{d_{\theta}}^{2}+\ldots+c_{1}^{2}+\left(c_{d_{\theta}%}^{2}+\ldots+c_{1}^{2}+c_{0}^{2}\right)^{2}}$
		$\displaystyle\leq\sqrt{2\left(c_{d_{\theta}}^{2}+\ldots+c_{1}^{2}+c_{0}^{2}%\right)^{2}}=\sqrt{2}\ \|\|\ \mathbf{c}_{\infty}\|\|^{2}$

and hence although $||\mathbf{c}||$ depends on the precision of the approximation, it is bounded by the fixed constant $\sqrt{2}||\ \mathbf{c}_{\infty}||^{2}$ .

As explained in [19, Page 68], heuristically the length of the shortest vector in $\mathcal{L}$ is approximately $\Delta\left(\mathcal{L}\right)^{\frac{1}{d_{\theta}+1}}$ , where $\Delta\left(\mathcal{L}\right)$ is the determinant of the lattice $\mathcal{L}_{k}$ . Thus, for a sufficiently large precision, the vector $\mathbf{c}$ is expected to be the shortest vector in the lattice, and of norm significantly smaller than the stated bound. In our case, $\Delta\left(\mathcal{L}\right)=\text{det}\left(A\right)=C=10^{k^{\prime}}$ ; and so if the minimal polynomial has coefficients of order $10^{n}$ , $k$ , $k^{\prime}$ are such that:

$\left(d_{\theta}+1\right)10^{2n}\leq 10^{k^{\prime}/\left(d_{\theta}+1\right)}$

then short vectors in $\mathcal{L}$ are a suitable candidates for the vector of coefficients.

When the imaginary part of $\alpha$ is not small, our strategy is to simultaneously minimise both real and imaginary parts of the polynomial, and hence the same method can be applied to the lattice generated by the columns of

$A_{k}=\begin{pmatrix}1&\ldots&0&0\\0&\ldots&0&0\\\vdots&\ddots&\vdots&\vdots\\0&\ldots&1&0\\\left[C\text{Re}\left(\alpha^{d_{\theta}}\right)\right]&\ldots&\left[C\text{Re%}\left(\alpha\right)\right]&\left[C\right]\\\left[C\text{Im}\left(\alpha^{d_{\theta}}\right)\right]&\ldots&\left[C\text{Im%}\left(\alpha\right)\right]&0\end{pmatrix}$

where $\text{Re}\left(x\right)$ and $\text{Im}\left(x\right)$ denote the real and imaginary parts of $x$ ; and $C=10^{k^{\prime}}$ where $k^{{}^{\prime}}<k$ , is such that

$\mid\left[C\cdot\text{Re}\left(\alpha^{i}\right)\right]-C\cdot\text{Re}\left(%\theta^{i}\right)\mid\ \leq\ 1$ and $\mid\left[C\cdot\text{Im}\left(\alpha^{i}\right)\right]-C\cdot\text{Im}\left(%\theta^{i}\right)\mid\ \leq\ 1$

for all $0\leq i\leq d_{\theta}$ .

Note that the above construction depends on $d_{\theta}$ , which we don’t a priori know. However, we can replace $d_{\theta}$ in the construction by any positive integer $d$ and search for short vectors in the corresponding lattice. The strategy to find $\mathbf{c}$ is as follows; we begin with a candidate $d$ for $d_{\theta}$ , starting with $d=2$ , and running through the natural numbers. If our candidate $d$ is equal to $d_{\theta}$ , then the norm of the shortest vector is typically much smaller than the expected bound, and we identify it as a possible vector of coefficients.

Coefficient Relations

Given minimal polynomials $\left(f_{1},\ldots,f_{n}\right)$ of $\left(x_{1},\ldots,x_{n}\right)$ , we want to find which combinations of roots correspond to points on the scheme.

We do this by searching for relations amongst our coefficients using lattice reduction. We show how a possible relation between $x_{1}$ and $x_{2}$ is computed. If $\alpha_{1},\alpha_{2}\in\mathbb{R}$ , we search for $b_{0},\ldots,b_{d_{1}}$ such that

$-b_{d_{1}}x_{2}=b_{d_{1}-1}x_{1}^{d_{1}-1}+\ldots+b_{1}x_{1}+b_{0}$

where $d_{1}$ is the degree of the minimal polynomial of $f_{1}$ . As in our search for minimal polynomials, such integers can be found by computing short vectors in the lattice generate by the columns of

$\begin{pmatrix}1&\ldots&0&0&0\\0&\ldots&0&0&0\\\vdots&\ddots&\vdots&\vdots&\vdots\\0&\ldots&0&1&0\\\left[C\alpha_{1}^{d_{1}-1}\right]&\ldots&\left[C\alpha_{1}\right]&\left[C%\alpha_{2}\right]&\left[C\right]\\\end{pmatrix}$

where $C=10^{k^{\prime}}$ and $k^{\prime}<k$ is chosen similarly as before. If the imaginary parts of $\alpha_{1},\alpha_{2}$ are not both small, we instead search for short vectors in the lattice generated by the columns of

$\begin{pmatrix}1&\ldots&0&0&0\\0&\ldots&0&0&0\\\vdots&\ddots&\vdots&\vdots&\vdots\\0&\ldots&0&1&0\\\left[C\text{Re}\left(\alpha_{1}^{d_{1}-1}\right)\right]&\ldots&\left[C\text{%Re}\left(\alpha_{1}\right)\right]&\left[C\text{Re}\left(\alpha_{2}\right)%\right]&\left[C\right]\\\left[C\text{Im}\left(\alpha_{1}^{d_{1}-1}\right)\right]&\ldots&\left[C\text{%Im}\left(\alpha_{1}\right)\right]&\left[C\text{Im}\left(\alpha_{2}\right)%\right]&0\end{pmatrix}$

In all of our examples, for a given orbit of cubics, we were able to express all coefficients of the defining equations in terms of one fixed coefficient. Using these relations and the minimal polynomials, we verify their correctness by computing symbolically the roots and the corresponding points on the scheme, and verifying that these are solutions of the system of equations.

4.2. Continued Fractions and Minimal Polynomials

An alternative method for determining minimal polynomials is to use the theory of continued fractions. Let $y_{1},...,y_{d}$ be complex approximations to the roots of a rational monic polynomial of degree $d$ . Using Vieta’s formulae [13], we can construct a monic polynomial of degree $d$ whose roots are the $y_{i}$ . This polynomial will be an approximation to the original polynomial, and so it suffices to recognise the coefficients as rational numbers. The error propagation in this process is fairly tame, to first order, the relative error in the estimates is magnified by a factor of $d-n$ in computing the coefficients. To see this, let the relative errors in each $y_{i}$ be $\delta_{i}$ , then the absolute error in the coefficient of $x^{n}$ is the following.

\sum_{1\leq i_{1}\ldots i_{d-n}\leq n}y_{i_{1}}(1+\delta_{i_{1}})\ldots y_{i_{%d-n}}(1+\delta_{i_{d-n}})-\sum_{1\leq i_{1}\ldots i_{d-n}\leq n}y_{i_{1}}%\ldots y_{i_{d-n}}

=\sum_{i}\sum_{\begin{subarray}{c}1\leq j_{1},\ldots,j_{d-n-1}\leq d\\j_{k}\neq i\end{subarray}}y_{j_{1}}\ldots y_{j_{d-n-1}}y_{j}\delta_{j}\leq%\left(\sum_{i}\sum_{\begin{subarray}{c}1\leq j_{1},\ldots,j_{d-n-1}\leq d\\j_{k}\neq i\end{subarray}}y_{j_{1}}\ldots y_{j_{d-n-1}}y_{j}\right)\max_{i}{%\delta_{i}}

The double sum is then the coefficient counted $d-n$ times, once for each choice of $j$ .Unfortunately, truncation errors become an issue if there are roots of extreme sizes, so this method tends to work best when roots are all of a similar order of magnitude.

We focus now on recognising the coefficients as rationals. This method is standard in many undergraduate lecture notes, including [20]. Given a real number, $x$ , the “best” rational approximation comes from the continued fraction expansion. In this case, $x$ is a close approximation to a rational number, $q$ . The continued fraction for $q$ terminates, and so the continued fraction expansion for $x$ will, at some point, have a very small error, corresponding to the end point of the continued fraction. If $q=\frac{a}{b}$ and $x-q=\varepsilon$ , the error at this step is approximately $b^{2}\varepsilon$ , so for any rational, if $x$ is known with enough precision, the error will be below any given bound.

It is possible to factorise the resulting polynomial, and so compute Galois orbits. Our main use of this is to identify small orbits where lattice reduction techniques can be applied. As before, the correctness can be verified by computing symbolically the roots.

5. Example 1: Fermat Quartic

We work with the following affine model of the Fermat quartic

$X:x^{4}-y^{3}-4y=0$

to ensure that a flex is in the necessary configuration.The $3$ -torsion points on the Jacobian $J$ of $X$ are effective divisor $D$ , where $3D$ is the zero divisor of $h$ ,

$h=a_{1}y^{2}+a_{2}xy+a_{3}y+a_{4}x^{3}+a_{5}x^{2}+a_{6}x+a_{7}$ ,

for some algebraic numbers $a_{1},\ldots,a_{6}$ .

Following the method described in Section 2, we arrive at the following ten equations:

	$\displaystyle e_{1}$	$\displaystyle=4a_{2}^{2}a_{6}+a_{6}^{3}+8a_{6}^{2}+16a_{6}-a_{9}^{3}a_{10},$
	$\displaystyle e_{2}$	$\displaystyle=8a_{1}a_{2}a_{6}+4a_{2}^{2}a_{5}+3a_{5}a_{6}^{2}+16a_{5}a_{6}+16%a_{5}-3a_{8}a_{9}^{2}a_{10},$
	$\displaystyle e_{3}$	$\displaystyle=4a_{1}^{2}a_{6}+8a_{1}a_{2}a_{5}+4a_{2}^{2}a_{4}+3a_{4}a_{6}^{2}%+16a_{4}a_{6}+16a_{4}+3a_{5}^{2}a_{6}$
		$\displaystyle+8a_{5}^{2}-3a_{7}a_{9}^{2}a_{10}-3a_{8}^{2}a_{9}a_{10},$
	$\displaystyle e_{4}$	$\displaystyle=4a_{1}^{2}a_{5}+8a_{1}a_{2}a_{4}+4a_{2}^{2}a_{3}+3a_{3}a_{6}^{2}%+16a_{3}a_{6}+16a_{3}+6a_{4}a_{5}a_{6}$
		$\displaystyle+16a_{4}a_{5}+a_{5}^{3}-6a_{7}a_{8}a_{9}a_{10}-a_{8}^{3}a_{10}-3a%_{9}^{2}a_{10},$
	$\displaystyle e_{5}$	$\displaystyle=4a_{1}^{2}a_{4}+8a_{1}a_{2}a_{3}+a_{2}^{3}+3a_{2}a_{6}+4a_{2}+6a%_{3}a_{5}a_{6}+16a_{3}a_{5}+3a_{4}^{2}a_{6}+8a_{4}^{2}+3a_{4}a_{5}^{2}$
		$\displaystyle-3a_{7}^{2}a_{9}a_{1}-3a_{7}a_{8}^{2}a_{10}-6a_{8}a_{9}a_{10},$
	$\displaystyle e_{6}$	$\displaystyle=4a_{1}^{2}a_{3}+3a_{1}a_{2}^{2}+3a_{1}a_{6}+4a_{1}+3a_{2}a_{5}+6%a_{3}a_{4}a_{6}+16a_{3}a_{4}+3a_{3}a_{5}^{2}+3a_{4}^{2}a_{5}-$
		$\displaystyle 3a_{7}^{2}a_{8}a_{10}-6a_{7}a_{9}a_{10}-3a_{8}^{2}a_{10},$
	$\displaystyle e_{7}$	$\displaystyle=3a_{1}^{2}a_{2}+3a_{1}a_{5}+3a_{2}a_{4}+3a_{3}^{2}a_{6}+8a_{3}^{%2}+6a_{3}a_{4}a_{5}+a_{4}^{3}-a_{7}^{3}a_{10}-6a_{7}a_{8}a_{10}-3a_{9}a_{10},$
	$\displaystyle e_{8}$	$\displaystyle=a_{1}^{3}+3a_{1}a_{4}+3a_{2}a_{3}+3a_{3}^{2}a_{5}+3a_{3}a_{4}^{2%}-3a_{7}^{2}a_{10}-3a_{8}a_{10},$
	$\displaystyle e_{9}$	$\displaystyle=3a_{1}a_{3}+3a_{3}^{2}a_{4}-3a_{7}a_{10}-1,$
	$\displaystyle e_{10}$	$\displaystyle=a_{3}^{3}-a_{10}.$

We also require an additional eleventh equation:

$e_{11}=a_{11}r+1$

where $r$ is the resultant of two polynomials, as previously described.

Using the method described in the previous sections, we compute two orbits of solutions. We state our orbits as follows: we give the minimal polynomial $f$ of $a_{1}$ , and expressions $s_{2},\ldots,s_{6}$ for $a_{2},\ldots,a_{6}$ in terms of $a_{1}$ .

	$\displaystyle f=x^{8}-(1/6)x^{4}-1/432$
	$\displaystyle a_{1}=u$
	$\displaystyle a_{2}=72u^{6}-14u^{2}$
	$\displaystyle a_{3}=(1/864)(u^{7}-36u^{3})\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ %\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$
	$\displaystyle a_{4}=0$
	$\displaystyle a_{5}=0$
	$\displaystyle a_{6}=0$

	$\displaystyle f=x^{8}-8x^{7}+28x^{6}-56x^{5}+52x^{4}+16x^{3}-80x^{2}+64x-44$
	$\displaystyle a_{1}=u$
	$\displaystyle a_{2}=-2u$
	$\displaystyle a_{3}=(1/108)(-u^{7}+7u^{6}-21u^{5}+35u^{4}-26u^{3}-6u^{2}-34u+4%6)\ \ \ \ \ \$
	$\displaystyle a_{4}=(1/18)(u^{7}-7u^{6}+21u^{5}-35u^{4}+26u^{3}+6u^{2}-2u+26)$
	$\displaystyle a_{5}=(1/9)(-u^{7}+7u^{6}-21u^{5}+35u^{4}-26u^{3}-6u^{2}+20u-26)$
	$\displaystyle a_{6}=(1/27)(2u^{7}-14u^{6}+42u^{5}-70u^{4}+52u^{3}+12u^{2}-40u+%16)$

Another orbit of $3$ -torsion points can be obtained by working on the simpler scheme, where we assume that our cubics are of the form

$h=-xy+a_{1}y+a_{2}x^{3}+a_{3}x^{2}+a_{4}x+a_{5}$ ,

and derive equations for this scheme by eliminating variables as in the previous case. This scheme has degree $8$ and we find that its points form a single orbit. To describe our orbit, we give a minimal polynomial $f$ for $a_{3}$ and expression for the other coefficients in terms of $a_{3}$ .

	$\displaystyle f=x^{8}-(1/6)x^{4}-1/432$
	$\displaystyle a_{1}=0$
	$\displaystyle a_{2}=0$
	$\displaystyle a_{3}=u$
	$\displaystyle a_{4}=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ %\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ %\ \ \ \$
	$\displaystyle a_{5}=72u^{6}-14u^{2}$

We verify that the three orbits above generate the entire subgroup $J[3]\cong\left(\mathbb{Z}/3\mathbb{Z}\right)^{6}$ as follows. The cubics above are all defined over the degree $32$ number field $K$ given by

$x^{32}-8x^{31}+28x^{30}-64x^{29}+130x^{28}-216x^{27}+120x^{26}+420x^{25}-1058x%^{24}+1032x^{23}-516x^{22}-128x^{21}+3494x^{20}-15256x^{19}+39888x^{18}-76812x%^{17}+119719x^{16}-158376x^{15}+182304x^{14}-185360x^{13}+168338x^{12}-137584x%^{11}+101508x^{10}-67596x^{9}+40588x^{8}-21912x^{7}+10536x^{6}-4440x^{5}+1612x%^{4}-488x^{3}+112x^{2}-16x+1$ .

Let $H$ be the finite subgroup of $J$ corresponding to the points in above orbits. In the ring of integers of $K$ , $\mathcal{O}_{K}$ , there is a prime ideal $\mathfrak{p}$ of norm $121$ , and since $11$ is a prime of good reduction for the curve, reduction modulo $\mathfrak{p}$ induces an injection on torsion [14]. The image of $H$ under the reduction map

$r_{\mathfrak{p}}:J\left(K\right)\longrightarrow J\left(\mathbb{F}_{\mathfrak{p%}}\right)$

is $\left(\mathbb{Z}/3\mathbb{Z}\right)^{6}$ , and hence $H$ is necessarily the entire $3$ -torsion subgroup $J[3]$ .

6. Example 2: Klein Quartic

We work with the classical affine model of the Klein quartic

$X:x^{3}y+y^{3}+x=0$ .

and compute $a_{1},\ldots,a_{6}$ such that the cubic defined by

$h=-y^{2}+a_{1}xy+a_{2}y+a_{3}x^{3}+a_{4}x^{2}+a_{5}x+a_{6}$

gives a $3$ -torsion element. Then proceeding as before, we compute a system of $11$ equations whose solutions correspond to our coefficients:

	$\displaystyle e_{1}$	$\displaystyle=-a_{6}^{3}+a_{7}a_{10}^{3}$
	$\displaystyle e_{2}$	$\displaystyle=a_{2}^{3}+3a_{2}a_{6}-3a_{5}a_{6}^{2}+3a_{7}a_{9}a_{10}^{2}$
	$\displaystyle e_{3}$	$\displaystyle=3a_{1}a_{2}^{2}+3a_{1}a_{6}+3a_{2}a_{5}-3a_{4}a_{6}^{2}-3a_{5}^{%2}a_{6}+3a_{7}a_{8}a_{10}^{2}+3a_{7}a_{9}^{2}a_{10}+1$
	$\displaystyle e_{4}$	$\displaystyle=3a_{1}^{2}a_{2}+3a_{1}a_{5}-a_{2}^{2}a_{6}+3a_{2}a_{4}-3a_{3}a_{%6}^{2}-6a_{4}a_{5}a_{6}-a_{5}^{3}-2a_{6}^{2}+6a_{7}a_{8}a_{9}a_{10}$
		$\displaystyle+a_{7}a_{9}^{3}+3a_{7}a_{10}^{2}$
	$\displaystyle e_{5}$	$\displaystyle=a_{1}^{3}-2a_{1}a_{2}a_{6}+3a_{1}a_{4}-a_{2}^{2}a_{5}+3a_{2}a_{3%}+a_{2}-6a_{3}a_{5}a_{6}-3a_{4}^{2}a_{6}-3a_{4}a_{5}^{2}-4a_{5}a_{6}$
		$\displaystyle+3a_{7}a_{8}^{2}a_{10}+3a_{7}a_{8}a_{9}^{2}+6a_{7}a_{9}a_{10}$
	$\displaystyle e_{6}$	$\displaystyle=-a_{1}^{2}a_{6}-2a_{1}a_{2}a_{5}+3a_{1}a_{3}+a_{1}-a_{2}^{2}a_{4%}-6a_{3}a_{4}a_{6}-3a_{3}a_{5}^{2}-3a_{4}^{2}a_{5}-4a_{4}a_{6}-2a_{5}^{2}+$
		$\displaystyle 3a_{7}a_{8}^{2}a_{9}+6a_{7}a_{8}a_{10}+3a_{7}a_{9}^{2}$
	$\displaystyle e_{7}$	$\displaystyle=-a_{1}^{2}a_{5}-2a_{1}a_{2}a_{4}-a_{2}^{2}a_{3}-3a_{3}^{2}a_{6}-%6a_{3}a_{4}a_{5}-4a_{3}a_{6}-a_{4}^{3}-4a_{4}a_{5}-a_{6}+a_{7}a_{8}^{3}$
		$\displaystyle+6a_{7}a_{8}a_{9}+3a_{7}a_{10}$
	$\displaystyle e_{8}$	$\displaystyle=-a_{1}^{2}a_{4}-2a_{1}a_{2}a_{3}-3a_{3}^{2}a_{5}-3a_{3}a_{4}^{2}%-4a_{3}a_{5}-2a_{4}^{2}-a_{5}+3a_{7}a_{8}^{2}+3a_{7}a_{9}$
	$\displaystyle e_{9}$	$\displaystyle=-a_{1}^{2}a_{3}-3a_{3}^{2}a_{4}-4a_{3}a_{4}-a_{4}+3a_{7}a_{8}$
	$\displaystyle e_{10}$	$\displaystyle=-a_{3}^{3}-2a_{3}^{2}-a_{3}+a_{7}$

and $e_{11}$ is of the same form as before.

We computed two orbits of $3-$ torsion points, which were sufficient to generate the entire group. Below, we give the minimal polynomial of $a_{1}$ and expressions for $a_{2},\ldots,a_{6}$ in terms of $a_{1}$ .

	$\displaystyle f=x^{24}-8x^{23}+26x^{22}-41x^{21}-56x^{20}+574x^{19}-1155x^{18}%-694x^{17}+5853x^{16}-8258x^{15}$
	$\displaystyle+3219x^{14}+10360x^{13}-50995x^{12}+100695x^{11}+2805x^{10}-29296%9x^{9}+317510x^{8}$
	$\displaystyle+129778x^{7}-412412x^{6}+184030x^{5}+86352x^{4}-109108x^{3}+44589%x^{2}-10096x+1009$
	$\displaystyle a_{1}=u$
	$\displaystyle a_{2}=(-435951209630782598061899622055031741081043u^{23}+3571381%032078647937662964093936968549559816u^{22}$
	$\displaystyle-11675186943937772871765833623109150981124795u^{21}+1799441303685%3208659852788815336903508075466u^{20}+$
	$\displaystyle 25803105134858601596505401339502379186400075u^{19}-2591605839745%55706565515262686720818548120485u^{18}$
	$\displaystyle+523778969657186662425923810713338161891078304u^{17}+348072907551%789435809107520565847036765988817u^{16}$
	$\displaystyle-2730404500005887076839604167477670574091542272u^{15}+35954825934%08112395894126217940978060778347035u^{14}$
	$\displaystyle-986188455465036555220598727240139337437891836u^{13}-485055876725%8071782161957415352370889563689816u^{12}$
	$\displaystyle+22701694079780912714753691113809320814697165394u^{11}-4516599097%1324005401367325287504349654210544902u^{10}$
	$\displaystyle-4350970427615079664592122069492592494777616887u^{9}+139851370684%018908240276639991512957958324069749u^{8}-$
	$\displaystyle 136739143426559138092316479794678325514475238440u^{7}-8536629198%4978396809280557614820969636726185049u^{6}+$
	$\displaystyle 189680256334508145759051666398055695840530757217u^{5}-5526139234%5239867305623968203706680140884876328u^{4}-$
	$\displaystyle 55075392536259947627007300122250664442993817996u^{3}+43290107969%201394199156213749516811789886641122u^{2}-$
	$\displaystyle 11948407690343313517365897079962014827888771595u+$
	$\displaystyle 1400989280495029579140060998705568762225219478)/5751849267457435%38665843780338528325370639313$
	$\displaystyle a_{3}=(10851359474294552058287984884461501020590057u^{23}-817190%38025536173937233179495793012162624266u^{22}$
	$\displaystyle+245377679459405721202298359482579825396667854u^{21}-338782720559%068097616753930864099098357758880u^{20}-$
	$\displaystyle 746493635324089924940708652642792473871209878u^{19}+585650062725%6709818900701760050361512308331835u^{18}$
	$\displaystyle-9899483813483103730374585951499766067746283829u^{17}-11589240558%418811901896070360550192501326383175u^{16}$
	$\displaystyle+57667369707061295712529533694255114907889339378u^{15}-6426993134%2460213537224212870561356550003984762u^{14}$
	$\displaystyle+8189889324881641411485597554710538269916951576u^{13}+11264648049%4335948383678069924433782070221157687u^{12}$
	$\displaystyle-497333746503947924082424699830398428404637513659u^{11}+869602607%053148613894334678322697433477379785842u^{10}$
	$\displaystyle+389140661553860741469400461736540942505257041012u^{9}-2948632711%463800976952466586090926921102095439915u^{8}$
	$\displaystyle+2139040783064833689431294629146584522275096629872u^{7}+227884808%6018478613013512248350540208510091260783u^{6}$
	$\displaystyle-3359224506585421618658709914024059975644960846209u^{5}+395760548%753092675180756922943039130586047866540u^{4}$
	$\displaystyle+1041608781211584080916917919674853216647175438909u^{3}-540164732%350553628603894902518822208600407608198u^{2}$
	$\displaystyle+168611615779118510909413482566618928175609369735u-$
	$\displaystyle 98416959483292631610131442805239714659254664127)/776499651106753%77719888910345701323925036307255$
	$\displaystyle a_{4}=(-4280989444220521273355279146893946493323224u^{23}+246394%79058711600676151318566019778263175555u^{22}$
	$\displaystyle-50485258220276792422340072320594133380745252u^{21}+3092358545238%4024083555134385652914240717380u^{20}$
	$\displaystyle+369423365960467057878944005592714249638226513u^{19}-165096863468%0527086639304585737871266965856330u^{18}$
	$\displaystyle+738501888882662895281074066428614372043526721u^{17}+671923795150%9837495058602443361228608156262712u^{16}$
	$\displaystyle-10548957760828800311184917125527530990893007084u^{15}+2168125216%720129517132739031863051638507552569u^{14}$
	$\displaystyle+4026507937684564168610386651329663488170878244u^{13}-33158336375%748733632461180590108006514265100451u^{12}$
	$\displaystyle+133637696782723810449041484760347867993674512833u^{11}-907375550%33670455033695709188896317975260562829u^{10}$
	$\displaystyle-377688896859637892313328913643364132917720792735u^{9}+4919015886%92832567689280486837874920357228701469u^{8}$
	$\displaystyle+297976029762786485325839530032278310021054468699u^{7}-4988486434%68620329349944578305346796225619927233u^{6}$
	$\displaystyle-19886042473720152793089204264539879829727399001u^{5}+20975357803%476830147175528026135198155461717127u^{4}$
	$\displaystyle+44781829656325113113779834055057941350014782838u^{3}+11220523427%4266884973910176728320702195587456234u^{2}$
	$\displaystyle-57466769816946581245715260299684504185894775922u-$
	$\displaystyle 4951133832496881111155518448009087122653328237)/2588332170355845%9239962970115233774641678769085$
	$\displaystyle a_{5}=(64672624496485254887769674249475137418437317u^{23}-485678%253783067121329174115525745954174622332u^{22}$
	$\displaystyle+1446858105729538443117884304295047358803487578u^{21}-19644140885%16181338426368967459248822043307617u^{20}$
	$\displaystyle-4532326603932548675323126511769456286250895459u^{19}+34857285117%255339129843631836731781005761922392u^{18}$
	$\displaystyle-57913040225154723934899044557721410711289619083u^{17}-7172502178%0697445983634475655422014556915035081u^{16}$
	$\displaystyle+342013102863593795926888669427091384215708051101u^{15}-372018515%858781494436249537857856162573039190031u^{14}$
	$\displaystyle+38518988612521747798642097388632708327825520870u^{13}+6833206443%00569286753582918685333030052822410904u^{12}$
	$\displaystyle-2970397482241197003218319717238841580709777646850u^{11}+50888467%75049677228073927340176029292630832884211u^{10}$
	$\displaystyle+2552854014890015141645854721302234484522422768968u^{9}-175644580%27063890239228046835218945544668191354686u^{8}+$
	$\displaystyle 12228030745762984509661653721081820434650658673556u^{7}+13729158%188932377522312895806683404051597636444383u^{6}-$
	$\displaystyle 20010176287626102917649998188221409158081985418464u^{5}+29201871%99720249325890543748558945060017654265590u^{4}+$
	$\displaystyle 6697716288602934383495117059686887992390102935500u^{3}-409749742%3913720638831271286382276506309116731849u^{2}+$
	$\displaystyle 1072361473607129792722376751912518716212791248650u$
	$\displaystyle-140537571390326774142168254703178467189717064628)/25883321703558%459239962970115233774641678769085$
	$\displaystyle a_{6}=(6023677266979226457200165086272886713906942u^{23}-3325596%4753029431649971530443586216025263516u^{22}+$
	$\displaystyle 49743084875179578611980774091455678344097670u^{21}+5571297626444%3586744246225556477420522677963u^{20}-$
	$\displaystyle 724706774550250490939043727386021186327225805u^{19}+237439466292%7561546767947263400803290191034247u^{18}+$
	$\displaystyle 603554635240505204370832490836018670740765548u^{17}-153707436395%97909868801523557223131530337132952u^{16}+$
	$\displaystyle 17511412634345062676424926408790855657793893075u^{15}+2018109932%3849093765711512691119322724031133019u^{14}-$
	$\displaystyle 50374845828844055245476612315031043335159939458u^{13}+6780720200%1509410710406320198969854893338968875u^{12}-$
	$\displaystyle 162635874991519313909979847644908542995046222065u^{11}-271701294%89972782010858032399852245676017068469u^{10}+$
	$\displaystyle 1016748221131521445736505446024631951651738622656u^{9}-104539877%6652722385272121676216102566602472234423u^{8}-$
	$\displaystyle 1621580886166598637426445029521256763885100478518u^{7}+276964653%7328602537764244216388282983330948674724u^{6}+$
	$\displaystyle 380499204117611513737183293098583683551288779273u^{5}-2407147944%497477746502019256665424886937109036216u^{4}+$
	$\displaystyle 788885838440605242375101797119590858573513442627u^{3}+5631874942%13492237644503207049700167772263489797u^{2}-$
	$\displaystyle 364190339047455474961286898795751878769633423734u+$
	$\displaystyle 48182833968966298128294670974310487182821926960)/776499651106753%77719888910345701323925036307255$

	$\displaystyle f$	$\displaystyle=1681x^{24}-5986x^{23}+22300x^{22}-39990x^{21}+52493x^{20}-69160x%^{19}+127435x^{18}-715541x^{17}+2982106x^{16}-8579387x^{15}$
		$\displaystyle+16843560x^{14}-21600425x^{13}+11307240x^{12}+27211590x^{11}-8816%4911x^{10}+142808549x^{9}-162840401x^{8}$
		$\displaystyle+135166869x^{7}-84018445x^{6}+43441328x^{5}-16680125x^{4}+2754739%x^{3}-71891x^{2}+83004x-18983$
	$\displaystyle a_{1}$	$\displaystyle=u$
	$\displaystyle a_{2}$	$\displaystyle=(508071131629120167504531502997193209451517119426255187640292227%5297812059962895646475519509466071148791u^{23}$
		$\displaystyle-1275369360745529573273858073364197673215484158935558187332665583%1351407883927919426061732744778964188151u^{22}$
		$\displaystyle+5406983211761491975878885672343475185877428860064556390604560592%0254189317594408019479510320011479331632u^{21}$
		$\displaystyle-6417181013970559331154968884539864751065431408721525199645339429%5100785428045746135683427184595158625737u^{20}$
		$\displaystyle+9181280256464886074007907005917128691870290902663802690919684969%8408351628788391203150122677920484230008u^{19}$
		$\displaystyle-1127841431988638863098299262107208936074011195697502846115210337%38271893099639293736870930764588320057506u^{18}$
		$\displaystyle+2669935744014363348806287971870405326921642317892881230161421711%55043722506161663604915778669773828347633u^{17}$
		$\displaystyle-1882434630026253110090507361593696475932312785977981282080964687%537281069003785439874439742965145563982151u^{16}$
		$\displaystyle+7037348216383439398255005275218849037350878365992615044777412069%900720944953368096124521037887282739732092u^{15}$
		$\displaystyle-1855881396173848845084009222845212620881641910305714901807107385%0987086335742763827370253050575813198524146u^{14}$
		$\displaystyle+3148299201943750283672176114817697785265558082184601569834268809%2458849196920763013751377697513421519775506u^{13}$
		$\displaystyle-3237490992750027269200510529813044350972063338570854002323781204%1007754588404370142018686585807055511680895u^{12}$
		$\displaystyle+3557691774964307479462551954760892593456597929664206865112501730%51079281457305855394233298739711155272619u^{11}$
		$\displaystyle+8259732919379673966523616252306420634302745845011089461556487014%3940026479112969448026127313140045973409904u^{10}$
		$\displaystyle-1801576503985308260721837287878522286323396795910901503467709068%10818215829832155807094288409714988689979556u^{9}+$
		$\displaystyle 2435800847817141549865626279649851009939587252160919200364598402%53065927598220207312391977205452274658388668u^{8}-$
		$\displaystyle 2377597385114143959382482544158998894605006205296369124382452337%84360496487088586670692225074624389340450754u^{7}+$
		$\displaystyle 1596943955983405715545569510355383212177931979505885571071873953%29156224600232620053563552890063887941835725u^{6}-$
		$\displaystyle 8600391180944382984099227227331539525383962193840573273107053571%1686629282342243034870290183101116842009764u^{5}+$
		$\displaystyle 3961219840926229694867224368966450817352018127751295782257421013%3699852603256862010435890773707310082876229u^{4}-$
		$\displaystyle 7429707948361218460675985566192592433627458250659525673161012946%787623695695907763802682362140849932504699u^{3}-$
		$\displaystyle 3303344515994141979285718483237240303085553677304572445801929959%81742168797733148853654492862067961911437u^{2}-$
		$\displaystyle 3778425577883191200249624394701203573240544943728725462713333589%7159740405773750398653020625508236955499u+$
		$\displaystyle 4958778712049824747116269250057687768908185059331979621765985125%2795712853244197552204673927671584747334)$
		$\displaystyle 2161498275682573022026195817125602128349794385606242898215815951%6725872132057987569950562665774724066953$
	$\displaystyle a_{3}$	$\displaystyle=(-12679536213946439318853414346136120497600366022692537930599186%21605641016810904866238021008440986740711u^{23}$
		$\displaystyle+5275373735623589195307457243762060865281425700352505105458982426%481720318122004342734954529439720012916u^{22}-$
		$\displaystyle 1755505556803198401044905118340749472465448572460868882238528507%6722424890771359537667612108115789516286u^{21}+$
		$\displaystyle 3607850424846609239989058588809709386378347266014971731359710585%4481920753785509504402589398466773809984u^{20}-$
		$\displaystyle 3794640858463179257339197667844310680734541607549157574893081775%2782479779199124812188792120685429509423u^{19}+$
		$\displaystyle 5825619319282596219097786055738105532155496231816235177001751087%3224764931395211715237402145426029985804u^{18}-$
		$\displaystyle 9520760827731143572115369185844447201700926104040724004997285990%1581903957772201656688626360027785665037u^{17}+$
		$\displaystyle 5653460827991583276468713693669007772209593511030861083236078283%03863859034761894583940459855975516785194u^{16}-$
		$\displaystyle 2476231504856247976047298453759515993343831083831653917785564323%136319966170797510904548844967133409657670u^{15}+$
		$\displaystyle 7125312232429473312047951375420296867035772620446711045133308422%259661885659812328712231767914046947934958u^{14}-$
		$\displaystyle 1410951227142093250403097076645375290377384455257356124007587874%9128624028151174987505867791907853260748076u^{13}+$
		$\displaystyle 1756017292934432454085281060013490348093828490582955434678716203%7635821366007680657414419368072805771438339u^{12}$
		$\displaystyle-8185431832605214591009680659652586245149536486491725589340235627%802890427221906509795978045372139654794230u^{11}$
		$\displaystyle-2495183341520121622921042828810020620595654652498937256585787199%8830738064658316396620449552045840774648931u^{10}$
		$\displaystyle+7647971486069283364510950716232372364148350907890716945390958580%4969957921961000380996735122563865728165733u^{9}-$
		$\displaystyle 1173957792347889292160046277414815189442309654431164029819295770%77829124619800405259020580076462879991849797u^{8}+$
		$\displaystyle 1282637375956060839362683810502044823763143473988534474790725950%36809193721366649969447441077094598291622074u^{7}-$
		$\displaystyle 9928532331903422929534487342577372831591029046762260137813418692%8070563896613305466138588259417348307050977u^{6}+$
		$\displaystyle 5436993055581044131320839999892074608200503955848282688897333208%6696898409314926047850098394085878279218025u^{5}-$
		$\displaystyle 2717893651980981613651131425258164885342034448134127672443030442%9620457718671912543389459540195418314978262u^{4}+$
		$\displaystyle 8936498262894022918521035969814361816088197835020755567538304524%904381640656220272920029079933030404195998u^{3}+$
		$\displaystyle 1760428367995620432793351383622329490017829575801229945978437636%67203382525903669069511374902748262707034u^{2}+$
		$\displaystyle 1235513326408374607124967655323646111212111399540235278655500894%84795930156394779589094505594735194585689u-$
		$\displaystyle 2478113038648745877190096486683460597889295418467049979072364513%86293632246561954505235415172247375021367)$
		$\displaystyle/1945348448114315719823576235413041915514814947045618608394234356%50532849188521888129555063991972516602577$
	$\displaystyle a_{4}$	$\displaystyle=(216021874826045255259120195504101020081645295735096177902077235%52126993177062018251821281784387073323023u^{23}-$
		$\displaystyle 5411864569951842186016469597901848698773308457908858913891929615%2554879292400289333451439352912231516049u^{22}+$
		$\displaystyle 2296323484547206780312287821783643593755123141599606019359210461%65742218176157986704758918174177036598027u^{21}-$
		$\displaystyle 2717142409383306567882784812518726655465665966496337725703350854%42481467613870243914878773892675607655031u^{20}+$
		$\displaystyle 3892400846715061789859915576685328968116194059713051221387290719%42997235822522833986887261769058574687206u^{19}-$
		$\displaystyle 4778096092405479823245604296011057072290974927226792594547717465%48049388515573721991902331874540454360677u^{18}+$
		$\displaystyle 1134307244791495073091508257061282838089035308203482251448828376%035701265351872142887357224173393587367217u^{17}-$
		$\displaystyle 7998241550184542723669282269744183912558735074840815505393916664%721266008981821312355998718933211562296046u^{16}+$
		$\displaystyle 2988435051370359364310231535633760722093513121673830331613706644%5444666720743574643185127746788378881399417u^{15}-$
		$\displaystyle 7876218536332316416604908623194805617242538309589132302392503969%8471564949121163272924901385013049687411942u^{14}+$
		$\displaystyle 1334828340085731965376356143492397351936573619304086126161073801%34767929957874016841359336191716810052372943u^{13}-$
		$\displaystyle 1370539607028841216966545384229913876400583276689827453860359134%92597685843425752919872849804140634643955270u^{12}+$
		$\displaystyle 1007732765399352112281231332865427721350305786463458957589706455%015055837197826332157186782126834563180874u^{11}+$
		$\displaystyle 3508150614696913575564818426235785051081810536268852468439572221%50401839155498648976140821847608132650413877u^{10}-$
		$\displaystyle 7638139739530900080182293149297389279327185140726300741827064787%99188355923066370269118304250507133175357855u^{9}+$
		$\displaystyle 1031649984516044068142490191656580092983268100075699233137397944%523070346671264968193662151018512880345885148u^{8}-$
		$\displaystyle 1006448713835171920508676798242363762206868954831955057736941294%417464520964723853766337832433285918952616800u^{7}+$
		$\displaystyle 6761216538662162508338570619696451050164010360472826020521086158%92432889649655824120715818859038851438719692u^{6}-$
		$\displaystyle 3653156306720622326546240410354755817061293551897903947739300356%66365242220829438972172664129492203081168695u^{5}+$
		$\displaystyle 1699283407803102280161653291225836915953141525018599588211903405%88839951126126425702399873705069760335867555u^{4}-$
		$\displaystyle 3302703151667362617481444452640544965834350455095245234759571790%5925904912242741639580360945517081829067445u^{3}-$
		$\displaystyle 5743989099745318961876529391811934579764144095733912977702259433%28270319549339393818609023946710634861218u^{2}-$
		$\displaystyle 7283961887395807853689242525478029686510111656985044981045262917%51071107199476419143522981427300087144081u+$
		$\displaystyle 2716648106989011986177066306518070177874476898230342248926278598%81305071186618314533095235468764334213720)$
		$\displaystyle/6484494827047719066078587451376806385049383156818728694647447855%0177616396173962709851687997324172200859$
	$\displaystyle a_{5}$	$\displaystyle=(389251092461903256679248649226977365667492071290372838181982016%80170886992904640592700404549963347199512u^{23}-$
		$\displaystyle 9730468700420251930163846303898433570238816176059736227842794098%2276163923129353032410251567051407236167u^{22}+$
		$\displaystyle 4134898813926743231301454264071788725918798832239081977319542580%71948111901480999775280191710173544246619u^{21}-$
		$\displaystyle 4878347253978336427359963399821566772607032373283918378246844258%17645677463132895183618864588299521039639u^{20}+$
		$\displaystyle 7011270139073202594296031801643467580647325066921266915553268285%09371238162600469908829030757269011344250u^{19}-$
		$\displaystyle 8590727659411750756484396573417334338054622570419879693154450735%40647832041592350614273857719119348322752u^{18}+$
		$\displaystyle 2043253808780106242301864341366623920410125557270465718105193925%121468028580643772742945785695630846250616u^{17}-$
		$\displaystyle 1440459109729876693302651404565898139613049566555005381185947657%3865203266087239366732093016561473185510947u^{16}+$
		$\displaystyle 5378209289358200522002905831824927375387193275717624301988828918%1093636039770292877616046638166811259966940u^{15}-$
		$\displaystyle 1417141918606095528801529749531814889907585929633927368863604365%31074827537157064474462945606682668819806198u^{14}+$
		$\displaystyle 2400473525765499093646017000360803907210214110164496592005719176%92510750285660599011803164564843128819228182u^{13}-$
		$\displaystyle 2464091478926679690813717105116636246422125609001724221059702239%33058322089161814282379652713411140663257935u^{12}+$
		$\displaystyle 1660383798745191426479235503825563309445604336426727713515545952%965583262056449973559241059379777205167680u^{11}+$
		$\displaystyle 6310307753473312862453741579718550529732552522361412259959446688%20543477829364676840720227312527178034180279u^{10}-$
		$\displaystyle 1373191303164392155556759433957455153048808084186727053603071678%098956093708891960388551820759614888919228798u^{9}+$
		$\displaystyle 1855055200398461753160983358264565827392481715440615201547027651%417550399650560016424464748539961643359247984u^{8}-$
		$\displaystyle 1810340272498862598842821857781647927048443171970743903664016426%775429620541753412488893145604156221308889189u^{7}+$
		$\displaystyle 1217447869444335701148390656031690559355995357559324353598131528%729900099121544745377288080145013432424025969u^{6}-$
		$\displaystyle 6598813846401719552688276670419019159300782353653045180083703811%72256477957510173898550948015864022299669197u^{5}+$
		$\displaystyle 3077907450656378642380889725083707773649703246756203908250608691%13848278840109786662025553536904120965133749u^{4}-$
		$\displaystyle 6082040786445341348065499786706472388760977110031535278351415480%2538810394027131443053153723537033134034705u^{3}-$
		$\displaystyle 1440520850906855000902365293707324495743371635969260100947138867%04776244290981162282467617876786399506180u^{2}-$
		$\displaystyle 1532941290409201639884641559299203829761919399394276833991478025%317804391841939489014019720099012115562420u+$
		$\displaystyle 5091221543201692684147998430645878273885017140621167474773429293%00007225772690649217374431742546708122792)$
		$\displaystyle/6484494827047719066078587451376806385049383156818728694647447855%0177616396173962709851687997324172200859$
	$\displaystyle a_{6}$	$\displaystyle=(106506330741605109288086249761949577032988062924119045234880864%834164563474865266431866700452076560904343u^{23}-$
		$\displaystyle 2686210672629222447317422908194733673249669462650123339975274224%87122752369679957389804773188189217975040u^{22}+$
		$\displaystyle 1137013977320349512141999486325566008661632741982097743275485317%535734426904814487385248439054139927647576u^{21}-$
		$\displaystyle 1359734100962858378220386886669806438734127279167679687329284208%050999992111901214621559610269099527897058u^{20}+$
		$\displaystyle 1945268188761932781006249424602429683919722293689218951137492143%206602562848341223292127451097837577427055u^{19}-$
		$\displaystyle 2393276829855285236300832480066408795592696499121108227777951838%724307749896235005081264299679364417582645u^{18}+$
		$\displaystyle 5636440654175695560827850572628403926238872648152811703245861566%191315889909066700423729024249080646852368u^{17}-$
		$\displaystyle 3953773984416060007673530537508577461331178240009048714319545609%1160379076939647113129281717859845031880493u^{16}+$
		$\displaystyle 1480197072478996808862243732925824794046917937073166241417604958%23122199250920962933822955373335955674707505u^{15}-$
		$\displaystyle 3909449032390166341798771185360585145462202874473958139656494542%25587344455995848059395553232190423750889714u^{14}+$
		$\displaystyle 6651699591715202443650248194705314527216546590014687171581079400%06910764868144221936179843690171495746867151u^{13}-$
		$\displaystyle 6881330229627556814681542082774658955050706467559944312761115161%32454231637364183867194761450409746615512057u^{12}+$
		$\displaystyle 1854701388347727343652707977124486913945778017845500029120703064%7792942254434253530679033647236399863545076u^{11}+$
		$\displaystyle 1727571258824689813458949909518392730988641427451135314145457582%390528356525372114445693870920521946183125968u^{10}-$
		$\displaystyle 3795550848736154518766013889265263853958641629778360330765324427%863117289700584168294491238114713048476467205u^{9}+$
		$\displaystyle 5156513047494830027630840938409945253924027656738811841516198563%878703922645799229211346774948819475873515647u^{8}-$
		$\displaystyle 5061011114440535922256772748616804053723528495298637982080885365%202535101405595927759267377757278846727374906u^{7}+$
		$\displaystyle 3433018553206648789129910985877222365044303100491891917848167429%332491962106076102926046515248888543454804836u^{6}-$
		$\displaystyle 1871323727407882096937532677095304025370320599490157355377914172%455447663767336731794094213574423386365089049u^{5}+$
		$\displaystyle 8751481206497582620607710467302073569314312429979228300332411808%73023842075954538819692035748582850095786175u^{4}-$
		$\displaystyle 1795184492246728531569040840881531830600942181027353208143379128%80840832807636239650130243559188053364442169u^{3}+$
		$\displaystyle 8674991122649621963177134855105027582856576655427398095162826312%37157815538071093053477697888868115188962u^{2}-$
		$\displaystyle 2891981098172373318053405115269016185141386151875879691697990113%793904535745021005613932188988315702933660u+$
		$\displaystyle 1150309215847546183632488145372536747955821185911367078149313606%638687782880520756923150814912414789090319)$
		$\displaystyle/1945348448114315719823576235413041915514814947045618608394234356%5053284918852188812955506399197251660257$

Both of our orbits are defined over the degree $48$ number field given by

$u^{48}-32u^{47}+492u^{46}-4838u^{45}+34136u^{44}-184200u^{43}+795782u^{42}-287%1488u^{41}+9027972u^{40}-25406816u^{39}+62125865u^{38}-110851470u^{37}+2020447%9u^{36}+896372004u^{35}-4721050815u^{34}+16331796445u^{33}-43395256901u^{32}+7%8805990590u^{31}-9062375006u^{30}-597966047368u^{29}+2489028250704u^{28}-53141%64476524u^{27}+4247868234838u^{26}+10286452763613u^{25}-38984213410246u^{24}+4%8194438473527u^{23}+34337356474920u^{22}-258320740169787u^{21}+493721560145034%u^{20}-279315253980219u^{19}-720711950420342u^{18}+1333301851341148u^{17}+6508%39726478139u^{16}-3536530808627111u^{15}+1476853576240844u^{14}+27732285279441%45u^{13}+2524087738848453u^{12}-10782439856962015u^{11}+1577760701045559u^{10}%+10769777804876901u^{9}+3034475537610752u^{8}-18205883402537385u^{7}+351890944%113304u^{6}+18574766492209602u^{5}-4386446310081687u^{4}-16132297574659372u^{3%}+15821424127424507u^{2}-6053794543417854u+920504949783049;$

and as in the previous example, we reduce modulo a prime of good reduction to verify that the two orbits listed above generate the entire $3-$ torsion subgroup $J\left[3\right]\cong\left(\mathbb{Z}/3\right)^{6}$ .

7. Local Conductor Exponents

A interesting application of these calculations is to compute local conductor exponents.

Let $C$ be any smooth, projective curve of genus $g$ defined over $\mathbb{Q}$ . The conductor of $C$ is an arithmetic invariant of the curve, defined as a product over the primes of bad reduction for the curve. The problem of computing conductors is essentially the problem of computing the local conductor exponent $n_{p}$ at each primes of bad reduction for the curve.

For elliptic curves, Tate’s algorithm [18] gives a complete resolution of the problem. This is also completely solved in the genus $2$ case by combining the cluster picture method described in [9] for $p\geq 3$ with the algorithm of Dokchitser-Doris in [10] for $p=2$ . In fact the results of [9] hold more generally for hyperelliptic curves and $p\geq 3$ . In [15], the method of [10] was generalised to compute part of $n_{2}$ for hyperelliptic curves of genus $3$ . Following [10], we explain how the $3-$ torsion subgroup of the Jacobian can be used to compute the wild part of $n_{2}$ .

Below, we give an overview of relevant definitions and results, see [22] for details.

To compute $n_{p}$ , we view $C$ as a smooth projective curve over $\mathbb{Q}_{p}$ , let $J$ be the Jacobian variety associated to $C$ and for a prime $l\neq p$ let $T=T_{l}J$ and $V=T\otimes_{\mathbb{Z}_{l}}\mathbb{Q}_{l}$ be the associated $l-$ adic Tate module and $l-$ adic representation respectively. Then $n_{p}$ is the Artin conductor of $V$ defined as follows

$n_{p}=\displaystyle\sum_{i=0}^{\infty}\frac{\text{codim}V^{G_{i}}}{[G_{0}:G_{i%}]}$

where $\{G_{i}\}_{i\geq-1}$ are the ramification subgroups $G=\text{Gal}\left(\bar{\mathbb{Q}}_{p}/\mathbb{Q}_{p}\right)$ in lower numbering, defined as

$\sigma\in G_{i}\iff v_{p}\left(\sigma\left(x\right)-x\right)\geq i+1\ \forall x%\in\mathbb{Z}_{p}$

The definition is independent of the choice of prime $l$ .

For computational purposes, it’s convenient to break up the above quantity into two parts. First, we observe that $G_{0}=I$ is the inertia subgroup, and define the ‘tame part’ of $n_{p}$ as

$n_{p,\text{tame}}=2g-\text{dim}V^{I}$ .

This is computable directly from the regular model of the curve over $\mathbb{Z}_{l}$ , as we can deduce the following invariants:

•
the abelian part $a$ , equal to the sum of the genera of all components of the model,
•
the toric part $t$ , equal to the number of loops in the dual graph of $C$ .

The tame part of the exponent is equal to $2g-2a-t$ , see [2, Chapter 9] for details. Regular models can be computed in principle by taking any model of the curve and performing repeated blowups until it becomes regular or by computing a $\Delta_{\nu}$ -regular model as in [8].

The second part of the conductor exponent is the ‘wild part’, defined as

$n_{p,\text{wild}}=\displaystyle\sum_{i=1}^{\infty}\frac{\text{codim}V^{G_{i}}}%{[G_{0}:G_{i}]}$ .

The above sum is, in fact, finite since $G_{i}=0$ for sufficiently large $i$ . Also, for $i\geq 1$ , $G_{i}$ is pro-p and $\text{codim}V^{G_{i}}=\text{codim}\bar{V}^{G_{i}}=\text{codim}J\left[l\right]^%{G_{i}}$ , hence the wild conductor exponent is:

$n_{p,\text{wild}}=\displaystyle\sum_{i=1}^{\infty}\frac{\text{codim}J\left[l%\right]^{G_{i}}}{[G_{0}:G_{i}]}$ .

Thus for a give curve explicitly knowing $J\left[2\right]$ and $J\left[3\right]$ is sufficient to determine all $n_{p,\text{wild}}$ .

Finding $n_{p,\text{wild}}$ for $p\geq 3$ is computationally a much simpler problem; one could apply the method described above to the $2$ -torsion subgroup, for which there is an explicit description.For a plane quartic $C$ , one could explicitly compute $J\left[2\right]$ and hence $n_{p,\text{wild}}$ for $p\geq 3$ using the bitangent lines to the curve, and for $p=2$ the wild conductor exponent can be determined using our explicit description of the $3-$ torsion subgroup.

8. Local Conductor Exponents at $2$ of the Fermat and Klein Quartics

We set $l=3$ in the previous section and compute $n_{2}$ in our examples. All of our computation are done in Magma, using the ‘Local Arithmetic Fields’ implementation [3].

8.1. The Fermat Quartic

Let $G=\operatorname{Gal}\left(\mathbb{Q}_{2}\left(f\right)/\mathbb{Q}_{2}\right)$ , where $f$ is the defining polynomial of $K$ from Section 5, and $\{G_{i}\}_{i\geq-1}$ the ramification groups in lower numbering. Using Magma, we find that $G$ is generated by five elements, $\sigma_{1},\ldots,\sigma_{5}$ , where $\sigma_{1},\sigma_{2}$ have order 3, and $\sigma_{3},\sigma_{4},\sigma_{5}$ have order 2. The ramification groups are as follows

	$\displaystyle G_{0}=G_{1}=<\sigma_{1},\sigma_{2},\sigma_{4},\sigma_{5}>,$
	$\displaystyle G_{2}=G_{3}=<\sigma_{4},\sigma_{5}>,$
	$\displaystyle G_{4}=G_{5}=G_{6}=G_{7}=<\sigma_{5}>,$
	$\displaystyle G_{n}=1\ \text{for all}\ n\geq 8,$

and $|G_{0}|=16,|G_{2}|=4,|G_{4}|=2$ . We compute the Galois action on the $3$ -torsion subgroup by computing the action of the generators on the roots of the minimal polynomials defining our $3$ -torsion orbits. We find

	$\displaystyle J[3]^{G_{0}}=J[3]^{G_{2}}=\{0\},$
	$\displaystyle J[3]^{G_{4}}\cong\left(\mathbb{Z}/3\mathbb{Z}\right)^{4}.$

From the formula, we compute the wild conductor exponent at $2$ to be

$n_{2,\text{wild}}=6+2\frac{6}{4}+4\frac{2}{8}=10$ .

Observe that as $J[3]^{G_{0}}=\{0\}$ , since the 3-adic Tate module maps onto the 3 torsion, $V^{G_{0}}=\{0\}$ and so the tame part of the conductor is 6. Combining this gives the conductor exponent at 2 as $n_{2}=16$ .In this example, we can verify our computation using the the modular structure of the Fermat quartic. The curve is isomorphic to $X_{0}(64)$ , and there are 3 cusp forms of this level and weight 2, two copies of the cusp form of level 32 and one of level 64. The conductors of these forms are the conductors of the corresponding elliptic curves arising in the Jacobian. Moreover, since the tame and wild parts of the conductor are preserved by the rational isogeny, the conductor exponent of the Fermat quartic is the sum of the 3 conductor exponents of the elliptic curves, which is $5+5+6=16$ in this case.

8.2. The Klein Quartic

We start by computing the conductor at 2. Since the number field defined by $f$ , as in Section 6, is unramified above 2, the ramification groups beyond $G_{0}$ are trivial. In particular, the wild conductor at 2 is trivial. This is to be expected, since the curve is known to have good reduction at 2, and so has trivial conductor at 2. Instead, we demonstrate the method at 7, the only bad prime for this curve. Let $G=\mathrm{Gal}(\mathbb{Q}_{7}(f)/\mathbb{Q}_{7})$ , where $f$ is the defining polynomial of $K$ from Section 6. We denote the ramification groups in the lower numbering by $\{G_{i}\}_{i\geq-1}$ . As $G$ has order 24, which is coprime to 7, $G_{1}$ is trivial since it is a $7$ -group. This shows the wild conductor is again, trivial.

As in the previous example, we compute that $J[3]^{G_{0}}=\{0\}$ , and so there are no non-trivial fixed points of the Tate module. This shows $n_{2,\mathrm{tame}}=6$ .

As in the case of the Fermat quartic, this can be deduced from the well understood theory of this curve. The Jacobian of the Klein quartic is isogenous to the cube of an elliptic curve with CM by $\mathbb{Q}(\sqrt{-7})$ [12]. This elliptic curve has conductor $7^{2}$ , and so $n_{7}=2\times 3=6$ . Moreover, as the conductor of the elliptic curve is all tame, the same is true of the Klein quartic, as noted above.

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