Genus 2 curve 11271.b.912951.1

• Label: 11271.b.912951.1
• Conductor: $11271$
• Discriminant: $-912951$
• 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

$y^2 + x^3y = 3x^3 + 7x^2 + 7x + 3$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(11271\)	\(=\)	\( 3 \cdot 13 \cdot 17^{2} \)
Discriminant:	\( \Delta \)	\(=\)	\(-912951\)	\(=\)	\( - 3^{5} \cdot 13 \cdot 17^{2} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(504\)	\(=\)	\( 2^{3} \cdot 3^{2} \cdot 7 \)
\( I_4 \)	\(=\)	\(9588\)	\(=\)	\( 2^{2} \cdot 3 \cdot 17 \cdot 47 \)
\( I_6 \)	\(=\)	\(1468800\)	\(=\)	\( 2^{7} \cdot 3^{3} \cdot 5^{2} \cdot 17 \)
\( I_{10} \)	\(=\)	\(3651804\)	\(=\)	\( 2^{2} \cdot 3^{5} \cdot 13 \cdot 17^{2} \)

Automorphism group

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

Rational points

Known points
\((1 : 0 : 0)\)	\((1 : -1 : 0)\)	\((-1 : 0 : 1)\)	\((-1 : 1 : 1)\)	\((-2 : 1 : 1)\)	\((1 : 4 : 1)\)
\((1 : -5 : 1)\)	\((-2 : 7 : 1)\)	\((-1 : -7 : 2)\)	\((-1 : 8 : 2)\)	\((1 : 23 : 2)\)	\((1 : -24 : 2)\)

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
\((-1 : 1 : 1) - (1 : -1 : 0)\)	\(z (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(z^3\)	\(0.456671\)	\(\infty\)
\((-1 : 1 : 1) + (1 : -5 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\)	\((x - z) (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-3xz^2 - 2z^3\)	\(0.016031\)	\(\infty\)

2-torsion field: 6.0.721344.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(2\)
Mordell-Weil rank:	\(2\)
2-Selmer rank:	\(2\)
Regulator:	\( 0.007288 \)
Real period:	\( 12.66344 \)
Tamagawa product:	\( 5 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 0.461473 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(3\)	\(1\)	\(5\)	\(5\)	\(( 1 - T )( 1 + 3 T + 3 T^{2} )\)
\(13\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 + 6 T + 13 T^{2} )\)
\(17\)	\(2\)	\(2\)	\(1\)	\(1 + 5 T + 17 T^{2}\)

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 \(\Q\).