Genus 2 curve 12544.e.50176.1

• Label: 12544.e.50176.1
• Conductor: $12544$
• Discriminant: $50176$
• Mordell-Weil group: \(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
• 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

$y^2 = x^5 + x^4 - 3x^3 - 2x^2 + 2x + 1$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(12544\)	\(=\)	\( 2^{8} \cdot 7^{2} \)
Discriminant:	\( \Delta \)	\(=\)	\(50176\)	\(=\)	\( 2^{10} \cdot 7^{2} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(166\)	\(=\)	\( 2 \cdot 83 \)
\( I_4 \)	\(=\)	\(532\)	\(=\)	\( 2^{2} \cdot 7 \cdot 19 \)
\( I_6 \)	\(=\)	\(28168\)	\(=\)	\( 2^{3} \cdot 7 \cdot 503 \)
\( I_{10} \)	\(=\)	\(196\)	\(=\)	\( 2^{2} \cdot 7^{2} \)

Automorphism group

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

Rational points

All points: \((1 : 0 : 0),\, (-1 : 0 : 1),\, (0 : -1 : 1),\, (0 : 1 : 1),\, (1 : 0 : 1),\, (5 : -6 : 4),\, (5 : 6 : 4)\)

Number of rational Weierstrass points: \(3\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)

Generator	$D_0$						Height	Order
\((0 : -1 : 1) - (1 : 0 : 0)\)	\(x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-z^3\)	\(0.206590\)	\(\infty\)
\((-1 : 0 : 1) + (1 : 0 : 1) - 2 \cdot(1 : 0 : 0)\)	\((x - z) (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0\)	\(2\)
\((1 : 0 : 1) - (1 : 0 : 0)\)	\(x - z\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0\)	\(2\)

2-torsion field: \(\Q(\zeta_{7})^+\)

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(1\)
Mordell-Weil rank:	\(1\)
2-Selmer rank:	\(3\)
Regulator:	\( 0.206590 \)
Real period:	\( 19.86190 \)
Tamagawa product:	\( 4 \)
Torsion order:	\( 4 \)
Leading coefficient:	\( 1.025818 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(8\)	\(10\)	\(4\)	\(1\)
\(7\)	\(2\)	\(2\)	\(1\)	\(1 + 4 T + 7 T^{2}\)

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?
\(2\)	2.240.1	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 \(\Q\).