Genus 2 curve 22112.b.353792.1

• Label: 22112.b.353792.1
• Conductor: $22112$
• Discriminant: $353792$
• 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

$y^2 + y = x^6 - 6x^4 - 9x^3 - 4x^2$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(22112\)	\(=\)	\( 2^{5} \cdot 691 \)
Discriminant:	\( \Delta \)	\(=\)	\(353792\)	\(=\)	\( 2^{9} \cdot 691 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(84\)	\(=\)	\( 2^{2} \cdot 3 \cdot 7 \)
\( I_4 \)	\(=\)	\(1113\)	\(=\)	\( 3 \cdot 7 \cdot 53 \)
\( I_6 \)	\(=\)	\(12381\)	\(=\)	\( 3 \cdot 4127 \)
\( I_{10} \)	\(=\)	\(44224\)	\(=\)	\( 2^{6} \cdot 691 \)

Automorphism group

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

Rational points

Known points
\((1 : -1 : 0)\)	\((1 : 1 : 0)\)	\((0 : 0 : 1)\)	\((-1 : 0 : 1)\)	\((0 : -1 : 1)\)	\((-1 : -1 : 1)\)
\((-1 : -3 : 2)\)	\((-1 : -5 : 2)\)	\((-2 : -8 : 3)\)	\((-3 : 9 : 2)\)	\((-3 : -17 : 2)\)	\((-2 : -19 : 3)\)
\((1 : -51 : 5)\)	\((1 : -74 : 5)\)

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
\((-1 : -3 : 2) + (-1 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\)	\((x + z) (2x + z)\)	\(=\)	\(0,\)	\(4y\)	\(=\)	\(-3xz^2 - 3z^3\)	\(0.220586\)	\(\infty\)
\((-1 : -1 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\)	\(x (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(xz^2\)	\(0.031601\)	\(\infty\)

2-torsion field: 6.2.1415168.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(2\)
Mordell-Weil rank:	\(2\)
2-Selmer rank:	\(2\)
Regulator:	\( 0.006891 \)
Real period:	\( 15.84805 \)
Tamagawa product:	\( 8 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 0.873731 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(5\)	\(9\)	\(8\)	\(1\)
\(691\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 + 38 T + 691 T^{2} )\)

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 \(\Q\).