Genus 2 curve 1996.a.31936.1

• Label: 1996.a.31936.1
• Conductor: $1996$
• Discriminant: $31936$
• Mordell-Weil group: \(\Z/{5}\Z\)
• 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

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

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(1996\)	\(=\)	\( 2^{2} \cdot 499 \)
Discriminant:	\( \Delta \)	\(=\)	\(31936\)	\(=\)	\( 2^{6} \cdot 499 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(1580\)	\(=\)	\( 2^{2} \cdot 5 \cdot 79 \)
\( I_4 \)	\(=\)	\(146065\)	\(=\)	\( 5 \cdot 131 \cdot 223 \)
\( I_6 \)	\(=\)	\(58442167\)	\(=\)	\( 7 \cdot 8348881 \)
\( I_{10} \)	\(=\)	\(-4087808\)	\(=\)	\( - 2^{13} \cdot 499 \)

Automorphism group

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

Rational points

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

Number of rational Weierstrass points: \(1\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

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

Generator	$D_0$						Height	Order
\(D_0 - 2 \cdot(1 : -1 : 0)\)	\(z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0\)	\(5\)

2-torsion field: 5.1.31936.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(0\)
Regulator:	\( 1 \)
Real period:	\( 13.67639 \)
Tamagawa product:	\( 1 \)
Torsion order:	\( 5 \)
Leading coefficient:	\( 0.547055 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(2\)	\(6\)	\(1\)	\(1 + T^{2}\)
\(499\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 - 5 T + 499 T^{2} )\)

Galois representations

The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) except those listed.

Prime \(\ell\)	mod-\(\ell\) image	Is torsion prime?
\(2\)	2.6.1	no
\(5\)	not computed	yes

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 \(\Q\).