Genus 2 curve 11011.b.77077.1

• Label: 11011.b.77077.1
• Conductor: $11011$
• Discriminant: $77077$
• 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

$y^2 + x^3y = x^5 + x^4 + 2x^3 + 6x^2 + 3x - 1$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(11011\)	\(=\)	\( 7 \cdot 11^{2} \cdot 13 \)
Discriminant:	\( \Delta \)	\(=\)	\(77077\)	\(=\)	\( 7^{2} \cdot 11^{2} \cdot 13 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(432\)	\(=\)	\( 2^{4} \cdot 3^{3} \)
\( I_4 \)	\(=\)	\(1428\)	\(=\)	\( 2^{2} \cdot 3 \cdot 7 \cdot 17 \)
\( I_6 \)	\(=\)	\(-631800\)	\(=\)	\( - 2^{3} \cdot 3^{5} \cdot 5^{2} \cdot 13 \)
\( I_{10} \)	\(=\)	\(308308\)	\(=\)	\( 2^{2} \cdot 7^{2} \cdot 11^{2} \cdot 13 \)

Automorphism group

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

Rational points

Known points
\((1 : 0 : 0)\)	\((1 : -1 : 0)\)	\((-1 : 0 : 1)\)	\((-1 : 1 : 1)\)	\((1 : 3 : 1)\)	\((-2 : 3 : 1)\)
\((1 : -4 : 1)\)	\((-2 : 5 : 1)\)	\((1 : 23 : 3)\)	\((1 : -24 : 3)\)	\((-5 : 59 : 2)\)	\((-5 : 66 : 2)\)

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
\((-5 : 66 : 2) + (-1 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\)	\((x + z) (2x + 5z)\)	\(=\)	\(0,\)	\(2y\)	\(=\)	\(-11xz^2 - 11z^3\)	\(0.254900\)	\(\infty\)
\((-2 : 3 : 1) + (1 : 3 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\)	\((x - z) (x + 2z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(3z^3\)	\(0.054000\)	\(\infty\)

2-torsion field: 6.2.100672.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(2\)
Mordell-Weil rank:	\(2\)
2-Selmer rank:	\(2\)
Regulator:	\( 0.013244 \)
Real period:	\( 17.65113 \)
Tamagawa product:	\( 2 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 0.467556 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(7\)	\(1\)	\(2\)	\(2\)	\(( 1 - T )( 1 + 4 T + 7 T^{2} )\)
\(11\)	\(2\)	\(2\)	\(1\)	\(1 + T^{2}\)
\(13\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 + 6 T + 13 T^{2} )\)

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?
\(2\)	2.15.2	no

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 \(\Q\).