Genus 2 curve 8128.b.520192.1

• Label: 8128.b.520192.1
• Conductor: $8128$
• Discriminant: $520192$
• Mordell-Weil group: \(\Z/{7}\Z\)
• 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

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

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(8128\)	\(=\)	\( 2^{6} \cdot 127 \)
Discriminant:	\( \Delta \)	\(=\)	\(520192\)	\(=\)	\( 2^{12} \cdot 127 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(4\)	\(=\)	\( 2^{2} \)
\( I_4 \)	\(=\)	\(61\)	\(=\)	\( 61 \)
\( I_6 \)	\(=\)	\(-1524\)	\(=\)	\( - 2^{2} \cdot 3 \cdot 127 \)
\( I_{10} \)	\(=\)	\(-2032\)	\(=\)	\( - 2^{4} \cdot 127 \)

Automorphism group

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

Rational points

All points: \((1 : -1 : 0),\, (1 : 1 : 0),\, (0 : 0 : 1)\)

Number of rational Weierstrass points: \(1\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z/{7}\Z\)

Generator	$D_0$						Height	Order
\(D_0 - 2 \cdot(1 : -1 : 0)\)	\(z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(x^3\)	\(0\)	\(7\)

2-torsion field: 5.1.130048.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(0\)
Regulator:	\( 1 \)
Real period:	\( 10.03424 \)
Tamagawa product:	\( 7 \)
Torsion order:	\( 7 \)
Leading coefficient:	\( 1.433463 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(6\)	\(12\)	\(7\)	\(1 - T\)
\(127\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 + 5 T + 127 T^{2} )\)

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?
\(2\)	2.6.1	no
\(7\)	not computed	yes

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 \(\Q\).