Genus 2 curve 3336.c.80064.1

• Label: 3336.c.80064.1
• Conductor: $3336$
• Discriminant: $-80064$
• Mordell-Weil group: \(\Z/{2}\Z \oplus \Z/{8}\Z\)
• 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

$y^2 + (x + 1)y = 6x^5 + 2x^4 + 2x^2 + x$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(3336\)	\(=\)	\( 2^{3} \cdot 3 \cdot 139 \)
Discriminant:	\( \Delta \)	\(=\)	\(-80064\)	\(=\)	\( - 2^{6} \cdot 3^{2} \cdot 139 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(1152\)	\(=\)	\( 2^{7} \cdot 3^{2} \)
\( I_4 \)	\(=\)	\(-1056\)	\(=\)	\( - 2^{5} \cdot 3 \cdot 11 \)
\( I_6 \)	\(=\)	\(-430020\)	\(=\)	\( - 2^{2} \cdot 3^{2} \cdot 5 \cdot 2389 \)
\( I_{10} \)	\(=\)	\(-320256\)	\(=\)	\( - 2^{8} \cdot 3^{2} \cdot 139 \)

Automorphism group

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

Rational points

All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (-1 : -2 : 2),\, (-1 : -9 : 3)\)

Number of rational Weierstrass points: \(3\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z/{2}\Z \oplus \Z/{8}\Z\)

Generator	$D_0$						Height	Order
\((-1 : -9 : 3) + (-1 : -2 : 2) - 2 \cdot(1 : 0 : 0)\)	\((2x + z) (3x + z)\)	\(=\)	\(0,\)	\(2y\)	\(=\)	\(-xz^2 - z^3\)	\(0\)	\(2\)
\((0 : 0 : 1) - (1 : 0 : 0)\)	\(x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0\)	\(8\)

2-torsion field: 3.1.139.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(2\)
Regulator:	\( 1 \)
Real period:	\( 14.16690 \)
Tamagawa product:	\( 16 \)
Torsion order:	\( 16 \)
Leading coefficient:	\( 0.885431 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(3\)	\(6\)	\(8\)	\(1 - T\)
\(3\)	\(1\)	\(2\)	\(2\)	\(( 1 + T )( 1 + 3 T^{2} )\)
\(139\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 - 12 T + 139 T^{2} )\)

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 \(\Q\).