Genus 2 curve 60916.a.243664.1

• Label: 60916.a.243664.1
• Conductor: $60916$
• Discriminant: $-243664$
• Mordell-Weil group: \(\Z \oplus \Z \oplus \Z\)
• 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

$y^2 + x^3y = 2x^3 - x^2 - 2x + 1$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(60916\)	\(=\)	\( 2^{2} \cdot 97 \cdot 157 \)
Discriminant:	\( \Delta \)	\(=\)	\(-243664\)	\(=\)	\( - 2^{4} \cdot 97 \cdot 157 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(72\)	\(=\)	\( 2^{3} \cdot 3^{2} \)
\( I_4 \)	\(=\)	\(933\)	\(=\)	\( 3 \cdot 311 \)
\( I_6 \)	\(=\)	\(50337\)	\(=\)	\( 3^{2} \cdot 7 \cdot 17 \cdot 47 \)
\( I_{10} \)	\(=\)	\(30458\)	\(=\)	\( 2 \cdot 97 \cdot 157 \)

Automorphism group

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

Rational points

Known points
\((1 : 0 : 0)\)	\((1 : -1 : 0)\)	\((-1 : 0 : 1)\)	\((0 : -1 : 1)\)	\((0 : 1 : 1)\)	\((1 : 0 : 1)\)
\((-1 : 1 : 1)\)	\((1 : -1 : 1)\)	\((1 : 0 : 2)\)	\((1 : -1 : 2)\)	\((2 : 1 : 1)\)	\((-2 : 3 : 1)\)
\((-2 : 5 : 1)\)	\((3 : 5 : 2)\)	\((2 : -9 : 1)\)	\((-2 : -27 : 3)\)	\((3 : -32 : 2)\)	\((-2 : 35 : 3)\)
\((-7 : -31625 : 30)\)	\((-7 : 31968 : 30)\)

Number of rational Weierstrass points: \(0\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z \oplus \Z \oplus \Z\)

Generator	$D_0$						Height	Order
\((0 : 1 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\)	\(x (x - z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-2xz^2 + z^3\)	\(0.457175\)	\(\infty\)
\((0 : -1 : 1) - (1 : -1 : 0)\)	\(z x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-z^3\)	\(0.301854\)	\(\infty\)
\((0 : -1 : 1) - (1 : 0 : 0)\)	\(z x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-x^3 - z^3\)	\(0.110995\)	\(\infty\)

2-torsion field: 6.4.3898624.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(3\)   (upper bound)
Mordell-Weil rank:	\(3\)
2-Selmer rank:	\(3\)
Regulator:	\( 0.014488 \)
Real period:	\( 16.65934 \)
Tamagawa product:	\( 3 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 0.724083 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(2\)	\(4\)	\(3\)	\(1 + 2 T + 2 T^{2}\)
\(97\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 + 12 T + 97 T^{2} )\)
\(157\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 + 2 T + 157 T^{2} )\)

Galois representations

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

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 \(\Q\).