Genus 2 curve 22131.a.199179.1

• Label: 22131.a.199179.1
• Conductor: $22131$
• Discriminant: $-199179$
• 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

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

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(22131\)	\(=\)	\( 3^{2} \cdot 2459 \)
Discriminant:	\( \Delta \)	\(=\)	\(-199179\)	\(=\)	\( - 3^{4} \cdot 2459 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(508\)	\(=\)	\( 2^{2} \cdot 127 \)
\( I_4 \)	\(=\)	\(73225\)	\(=\)	\( 5^{2} \cdot 29 \cdot 101 \)
\( I_6 \)	\(=\)	\(17430667\)	\(=\)	\( 17430667 \)
\( I_{10} \)	\(=\)	\(25494912\)	\(=\)	\( 2^{7} \cdot 3^{4} \cdot 2459 \)

Automorphism group

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

Rational points

Known points
\((1 : 1 : 0)\)	\((1 : -2 : 0)\)	\((0 : 0 : 1)\)	\((-1 : 0 : 1)\)	\((0 : -1 : 1)\)	\((-1 : 1 : 1)\)
\((1 : -1 : 1)\)	\((1 : -2 : 1)\)	\((1 : -3 : 2)\)	\((-2 : -5 : 1)\)	\((1 : -10 : 2)\)	\((-2 : 14 : 1)\)
\((4 : -233 : 9)\)	\((4 : -884 : 9)\)

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
\((0 : 0 : 1) + (1 : -2 : 1) - (1 : -2 : 0) - (1 : 1 : 0)\)	\(x (x - z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-2xz^2\)	\(0.097545\)	\(\infty\)
\((1 : -1 : 1) - (1 : -2 : 0)\)	\(z (x - z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(x^3 - 2z^3\)	\(0.066609\)	\(\infty\)

2-torsion field: 6.0.1416384.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(2\)
Mordell-Weil rank:	\(2\)
2-Selmer rank:	\(2\)
Regulator:	\( 0.006174 \)
Real period:	\( 18.61562 \)
Tamagawa product:	\( 5 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 0.574735 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(3\)	\(2\)	\(4\)	\(5\)	\(( 1 - T )( 1 + T )\)
\(2459\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 - 72 T + 2459 T^{2} )\)

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 \(\Q\).