Genus 2 curve 4264.a.699296.1

• Label: 4264.a.699296.1
• Conductor: $4264$
• Discriminant: $-699296$
• Mordell-Weil group: \(\Z/{10}\Z\)
• 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

$y^2 + (x + 1)y = -3x^6 + 8x^5 - 16x^4 + 18x^3 - 17x^2 + 8x - 4$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(4264\)	\(=\)	\( 2^{3} \cdot 13 \cdot 41 \)
Discriminant:	\( \Delta \)	\(=\)	\(-699296\)	\(=\)	\( - 2^{5} \cdot 13 \cdot 41^{2} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(9296\)	\(=\)	\( 2^{4} \cdot 7 \cdot 83 \)
\( I_4 \)	\(=\)	\(52144\)	\(=\)	\( 2^{4} \cdot 3259 \)
\( I_6 \)	\(=\)	\(158718244\)	\(=\)	\( 2^{2} \cdot 83 \cdot 478067 \)
\( I_{10} \)	\(=\)	\(2797184\)	\(=\)	\( 2^{7} \cdot 13 \cdot 41^{2} \)

Automorphism group

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

Rational points

This curve has no rational points.

Number of rational Weierstrass points: \(0\)

This curve is locally solvable except over $\R$.

Mordell-Weil group of the Jacobian

Group structure: \(\Z/{10}\Z\)

Generator	$D_0$						Height	Order
\(D_0 - D_\infty\)	\(2x^2 - 3xz + 4z^2\)	\(=\)	\(0,\)	\(4y\)	\(=\)	\(3xz^2 - 8z^3\)	\(0\)	\(10\)

2-torsion field: 8.0.44302336.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(2\)
Regulator:	\( 1 \)
Real period:	\( 4.514986 \)
Tamagawa product:	\( 10 \)
Torsion order:	\( 10 \)
Leading coefficient:	\( 0.902997 \)
Analytic order of Ш:	\( 2 \)   (rounded)
Order of Ш:	twice a square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(3\)	\(5\)	\(5\)	\(1 - T\)
\(13\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 + 6 T + 13 T^{2} )\)
\(41\)	\(1\)	\(2\)	\(2\)	\(( 1 - T )( 1 - 2 T + 41 T^{2} )\)

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?
\(2\)	2.45.1	yes
\(5\)	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 \(\Q\).