Genus 2 curve 6672.a.720576.1

• Label: 6672.a.720576.1
• Conductor: $6672$
• Discriminant: $-720576$
• Mordell-Weil group: \(\Z \oplus \Z/{2}\Z \oplus \Z/{4}\Z\)
• 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

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

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(6672\)	\(=\)	\( 2^{4} \cdot 3 \cdot 139 \)
Discriminant:	\( \Delta \)	\(=\)	\(-720576\)	\(=\)	\( - 2^{6} \cdot 3^{4} \cdot 139 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(728\)	\(=\)	\( 2^{3} \cdot 7 \cdot 13 \)
\( I_4 \)	\(=\)	\(15256\)	\(=\)	\( 2^{3} \cdot 1907 \)
\( I_6 \)	\(=\)	\(3297276\)	\(=\)	\( 2^{2} \cdot 3^{2} \cdot 91591 \)
\( I_{10} \)	\(=\)	\(-2882304\)	\(=\)	\( - 2^{8} \cdot 3^{4} \cdot 139 \)

Automorphism group

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

Rational points

All points
\((1 : 0 : 0)\)	\((0 : 0 : 1)\)	\((1 : 0 : 1)\)	\((-1 : -1 : 1)\)	\((1 : -1 : 1)\)	\((-1 : 2 : 1)\)
\((1 : -2 : 2)\)	\((2 : 2 : 1)\)	\((2 : -4 : 1)\)	\((25 : -2010 : 18)\)	\((25 : -6090 : 18)\)

Number of rational Weierstrass points: \(3\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z \oplus \Z/{2}\Z \oplus \Z/{4}\Z\)

Generator	$D_0$						Height	Order
\((-1 : -1 : 1) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)\)	\((x - z) (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-z^3\)	\(0.157524\)	\(\infty\)
\((0 : 0 : 1) + (1 : -2 : 2) - 2 \cdot(1 : 0 : 0)\)	\(x (2x - z)\)	\(=\)	\(0,\)	\(2y\)	\(=\)	\(-xz^2\)	\(0\)	\(2\)
\((1 : -1 : 1) + (2 : 2 : 1) - 2 \cdot(1 : 0 : 0)\)	\((x - 2z) (x - z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(3xz^2 - 4z^3\)	\(0\)	\(4\)

2-torsion field: 3.1.139.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(1\)
Mordell-Weil rank:	\(1\)
2-Selmer rank:	\(3\)
Regulator:	\( 0.157524 \)
Real period:	\( 15.89105 \)
Tamagawa product:	\( 16 \)
Torsion order:	\( 8 \)
Leading coefficient:	\( 0.625807 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(4\)	\(6\)	\(4\)	\(1 + T\)
\(3\)	\(1\)	\(4\)	\(4\)	\(( 1 - T )( 1 + 2 T + 3 T^{2} )\)
\(139\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 + 4 T + 139 T^{2} )\)

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?
\(2\)	2.120.3	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 \(\Q\).