Genus 2 curve 4312.a.551936.1

• Label: 4312.a.551936.1
• Conductor: $4312$
• Discriminant: $551936$
• Mordell-Weil group: \(\Z/{10}\Z\)
• Sato-Tate group: $\mathrm{USp}(4)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\R\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\Q\)
• \(\End(J) \otimes \Q\): \(\Q\)
• \(\overline{\Q}\)-simple: yes
• \(\mathrm{GL}_2\)-type: no

Minimal equation

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

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(4312\)	\(=\)	\( 2^{3} \cdot 7^{2} \cdot 11 \)
Discriminant:	\( \Delta \)	\(=\)	\(551936\)	\(=\)	\( 2^{10} \cdot 7^{2} \cdot 11 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(60\)	\(=\)	\( 2^{2} \cdot 3 \cdot 5 \)
\( I_4 \)	\(=\)	\(477\)	\(=\)	\( 3^{2} \cdot 53 \)
\( I_6 \)	\(=\)	\(-8865\)	\(=\)	\( - 3^{2} \cdot 5 \cdot 197 \)
\( I_{10} \)	\(=\)	\(-68992\)	\(=\)	\( - 2^{7} \cdot 7^{2} \cdot 11 \)

Automorphism group

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

Rational points

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

Number of rational Weierstrass points: \(2\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z/{10}\Z\)

Generator	$D_0$						Height	Order
\((0 : 0 : 1) - (1 : -1 : 0)\)	\(z x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0\)	\(10\)

2-torsion field: 4.0.2156.2

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(1\)
Regulator:	\( 1 \)
Real period:	\( 10.28259 \)
Tamagawa product:	\( 10 \)
Torsion order:	\( 10 \)
Leading coefficient:	\( 1.028259 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(3\)	\(10\)	\(10\)	\(1 - T\)
\(7\)	\(2\)	\(2\)	\(1\)	\(1 + T^{2}\)
\(11\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 - 2 T + 11 T^{2} )\)

Galois representations

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

Prime \(\ell\)	mod-\(\ell\) image
\(2\)	2.90.3

Sato-Tate group

\(\mathrm{ST}\)	\(\simeq\)	$\mathrm{USp}(4)$
\(\mathrm{ST}^0\)	\(\simeq\)	\(\mathrm{USp}(4)\)

Decomposition of the Jacobian

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

Endomorphisms of the Jacobian

Not of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)	\(\simeq\)	\(\Z\)
\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q\)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\R\)

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