Genus 2 curve 961.a.923521.1

• Label: 961.a.923521.1
• Conductor: $961$
• Discriminant: $923521$
• Mordell-Weil group: \(\Z/{5}\Z\)
• Sato-Tate group: $\mathrm{SU}(2)\times\mathrm{SU}(2)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\R \times \R\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathsf{RM}\)
• \(\End(J) \otimes \Q\): \(\mathsf{RM}\)
• \(\overline{\Q}\)-simple: yes
• \(\mathrm{GL}_2\)-type: yes

This is a model for the modular curve $X_0(31)$.

Minimal equation

$y^2 + (x^3 + x^2 + 1)y = -5x^4 + 4x^3 + 3x^2 - 2x - 3$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(961\)	\(=\)	\( 31^{2} \)
Discriminant:	\( \Delta \)	\(=\)	\(923521\)	\(=\)	\( 31^{4} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(4100\)	\(=\)	\( 2^{2} \cdot 5^{2} \cdot 41 \)
\( I_4 \)	\(=\)	\(78961\)	\(=\)	\( 281^{2} \)
\( I_6 \)	\(=\)	\(94151689\)	\(=\)	\( 94151689 \)
\( I_{10} \)	\(=\)	\(118210688\)	\(=\)	\( 2^{7} \cdot 31^{4} \)

Automorphism group

\(\mathrm{Aut}(X)\)	\(\simeq\)	$C_2$
\(\mathrm{Aut}(X_{\overline{\Q}})\)	\(\simeq\)	$C_2$

Rational points

All points: \((1 : 0 : 0),\, (1 : -1 : 0)\)

Number of rational Weierstrass points: \(0\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z/{5}\Z\)

Generator	$D_0$						Height	Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\)	\(x^2 - 3xz + 3z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-4xz^2 + 2z^3\)	\(0\)	\(5\)

2-torsion field: 3.1.31.1

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(0\)
Regulator:	\( 1 \)
Real period:	\( 2.246438 \)
Tamagawa product:	\( 5 \)
Torsion order:	\( 5 \)
Leading coefficient:	\( 0.449287 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(31\)	\(2\)	\(4\)	\(5\)	\(( 1 - T )^{2}\)

Galois representations

For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.

For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not all of $\GSp(4,\F_\ell)$.

Prime \(\ell\)	mod-\(\ell\) image	Is torsion prime?
\(2\)	2.120.2	no
\(3\)	3.72.2	no

Sato-Tate group

\(\mathrm{ST}\)	\(\simeq\)	$\mathrm{SU}(2)\times\mathrm{SU}(2)$
\(\mathrm{ST}^0\)	\(\simeq\)	\(\mathrm{SU}(2)\times\mathrm{SU}(2)\)

Decomposition of the Jacobian

Simple over \(\overline{\Q}\)

Endomorphisms of the Jacobian

Of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)	\(\simeq\)	\(\Z [\frac{1 + \sqrt{5}}{2}]\)
\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{5}) \)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\R \times \R\)

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).