Genus 2 curve 11529.a.726327.1

• Label: 11529.a.726327.1
• Conductor: $11529$
• Discriminant: $726327$
• Mordell-Weil group: \(\Z \oplus \Z\)
• 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

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

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(11529\)	\(=\)	\( 3^{3} \cdot 7 \cdot 61 \)
Discriminant:	\( \Delta \)	\(=\)	\(726327\)	\(=\)	\( 3^{5} \cdot 7^{2} \cdot 61 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(1572\)	\(=\)	\( 2^{2} \cdot 3 \cdot 131 \)
\( I_4 \)	\(=\)	\(162441\)	\(=\)	\( 3^{2} \cdot 18049 \)
\( I_6 \)	\(=\)	\(61761429\)	\(=\)	\( 3^{2} \cdot 6862381 \)
\( I_{10} \)	\(=\)	\(92969856\)	\(=\)	\( 2^{7} \cdot 3^{5} \cdot 7^{2} \cdot 61 \)

Automorphism group

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

Rational points

Known points
\((1 : 0 : 0)\)	\((1 : -1 : 0)\)	\((0 : 0 : 1)\)	\((0 : -1 : 1)\)	\((-1 : -1 : 1)\)	\((-1 : 2 : 1)\)
\((1 : 4 : 2)\)	\((2 : -4 : 1)\)	\((2 : -7 : 1)\)	\((3 : -9 : 1)\)	\((-3 : 11 : 1)\)	\((1 : -17 : 2)\)
\((-3 : 18 : 1)\)	\((3 : -22 : 1)\)

Number of rational Weierstrass points: \(0\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

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

Generator	$D_0$						Height	Order
\((-1 : -1 : 1) + (1 : 4 : 2) - (1 : -1 : 0) - (1 : 0 : 0)\)	\((x + z) (2x - z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(xz^2\)	\(0.110597\)	\(\infty\)
\((-1 : -1 : 1) - (1 : 0 : 0)\)	\(z (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-x^3 - 2z^3\)	\(0.038206\)	\(\infty\)

2-torsion field: 6.2.105408.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(2\)
Mordell-Weil rank:	\(2\)
2-Selmer rank:	\(2\)
Regulator:	\( 0.003877 \)
Real period:	\( 17.73093 \)
Tamagawa product:	\( 6 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 0.412496 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(3\)	\(3\)	\(5\)	\(3\)	\(1 + T + 3 T^{2}\)
\(7\)	\(1\)	\(2\)	\(2\)	\(( 1 + T )( 1 + 4 T + 7 T^{2} )\)
\(61\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 + 61 T^{2} )\)

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 \(\Q\).