Genus 2 curve 3515.a.333925.1

• Label: 3515.a.333925.1
• Conductor: $3515$
• Discriminant: $333925$
• Mordell-Weil group: \(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
• 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

$y^2 + (x^2 + 1)y = x^5 - 6x^3 + 2x^2 + 2x$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(3515\)	\(=\)	\( 5 \cdot 19 \cdot 37 \)
Discriminant:	\( \Delta \)	\(=\)	\(333925\)	\(=\)	\( 5^{2} \cdot 19^{2} \cdot 37 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(1144\)	\(=\)	\( 2^{3} \cdot 11 \cdot 13 \)
\( I_4 \)	\(=\)	\(47884\)	\(=\)	\( 2^{2} \cdot 11971 \)
\( I_6 \)	\(=\)	\(15303367\)	\(=\)	\( 31 \cdot 493657 \)
\( I_{10} \)	\(=\)	\(1335700\)	\(=\)	\( 2^{2} \cdot 5^{2} \cdot 19^{2} \cdot 37 \)

Automorphism group

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

Rational points

All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (1 : -1 : 1),\, (2 : -1 : 1),\, (2 : -4 : 1),\, (-1 : -34 : 4)\)

Number of rational Weierstrass points: \(3\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)

Generator	$D_0$						Height	Order
\((0 : -1 : 1) - (1 : 0 : 0)\)	\(x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-z^3\)	\(0.098008\)	\(\infty\)
\((-1 : -34 : 4) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)\)	\((x - z) (4x + z)\)	\(=\)	\(0,\)	\(8y\)	\(=\)	\(-3xz^2 - 5z^3\)	\(0\)	\(2\)
\((-1 : -34 : 4) - (1 : 0 : 0)\)	\(4x + z\)	\(=\)	\(0,\)	\(32y\)	\(=\)	\(-17z^3\)	\(0\)	\(2\)

2-torsion field: 3.3.148.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(1\)
Mordell-Weil rank:	\(1\)
2-Selmer rank:	\(3\)
Regulator:	\( 0.098008 \)
Real period:	\( 18.39250 \)
Tamagawa product:	\( 4 \)
Torsion order:	\( 4 \)
Leading coefficient:	\( 0.450656 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(5\)	\(1\)	\(2\)	\(2\)	\(( 1 - T )( 1 + 4 T + 5 T^{2} )\)
\(19\)	\(1\)	\(2\)	\(2\)	\(( 1 - T )( 1 + 6 T + 19 T^{2} )\)
\(37\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 - 10 T + 37 T^{2} )\)

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?
\(2\)	2.120.3	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 \(\Q\).