Genus 2 curve 88034.a.176068.1

• Label: 88034.a.176068.1
• Conductor: $88034$
• Discriminant: $176068$
• Mordell-Weil group: \(\Z \oplus \Z \oplus \Z\)
• 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

$y^2 + (x^3 + 1)y = 2x^5 - 10x^4 + 11x^3 - 2x^2 - x$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(88034\)	\(=\)	\( 2 \cdot 44017 \)
Discriminant:	\( \Delta \)	\(=\)	\(176068\)	\(=\)	\( 2^{2} \cdot 44017 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(3028\)	\(=\)	\( 2^{2} \cdot 757 \)
\( I_4 \)	\(=\)	\(146233\)	\(=\)	\( 257 \cdot 569 \)
\( I_6 \)	\(=\)	\(127602253\)	\(=\)	\( 131 \cdot 974063 \)
\( I_{10} \)	\(=\)	\(22536704\)	\(=\)	\( 2^{9} \cdot 44017 \)

Automorphism group

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

Rational points

Known points
\((1 : 0 : 0)\)	\((1 : -1 : 0)\)	\((0 : 0 : 1)\)	\((0 : -1 : 1)\)	\((1 : 0 : 1)\)	\((1 : -2 : 1)\)
\((2 : -3 : 1)\)	\((2 : -6 : 1)\)	\((1 : -12 : 3)\)	\((3 : -12 : 2)\)	\((1 : -16 : 3)\)	\((3 : -23 : 2)\)
\((1 : -24 : 4)\)	\((1 : -36 : 5)\)	\((1 : -41 : 4)\)	\((1 : -90 : 5)\)

Number of rational Weierstrass points: \(0\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

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

Generator	$D_0$						Height	Order
\((1 : -16 : 3) - (1 : 0 : 0)\)	\(z (3x - z)\)	\(=\)	\(0,\)	\(9y\)	\(=\)	\(-9x^3 - 5z^3\)	\(0.703530\)	\(\infty\)
\(2 \cdot(1 : -2 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\)	\((x - z)^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-2xz^2\)	\(0.585526\)	\(\infty\)
\((0 : 0 : 1) + (1 : -2 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\)	\(x (x - z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-2xz^2\)	\(0.057772\)	\(\infty\)

2-torsion field: 6.2.2817088.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(3\)   (upper bound)
Mordell-Weil rank:	\(3\)
2-Selmer rank:	\(3\)
Regulator:	\( 0.023187 \)
Real period:	\( 19.36559 \)
Tamagawa product:	\( 2 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 0.898085 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(1\)	\(2\)	\(2\)	\(( 1 + T )( 1 + T + 2 T^{2} )\)
\(44017\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 - 118 T + 44017 T^{2} )\)

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 \(\Q\).