Genus 2 curve 13689.c.41067.1

• Label: 13689.c.41067.1
• Conductor: $13689$
• Discriminant: $41067$
• Mordell-Weil group: \(\Z\)
• 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

$y^2 + (x^3 + x)y = x^5 - 6x^3 - 13x^2 - 6x + 9$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(13689\)	\(=\)	\( 3^{4} \cdot 13^{2} \)
Discriminant:	\( \Delta \)	\(=\)	\(41067\)	\(=\)	\( 3^{5} \cdot 13^{2} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(616\)	\(=\)	\( 2^{3} \cdot 7 \cdot 11 \)
\( I_4 \)	\(=\)	\(31720\)	\(=\)	\( 2^{3} \cdot 5 \cdot 13 \cdot 61 \)
\( I_6 \)	\(=\)	\(4631783\)	\(=\)	\( 13^{2} \cdot 27407 \)
\( I_{10} \)	\(=\)	\(-676\)	\(=\)	\( - 2^{2} \cdot 13^{2} \)

Automorphism group

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

Rational points

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

Number of rational Weierstrass points: \(0\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z\)

Generator	$D_0$						Height	Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\)	\(2x^2 + 7xz + 9z^2\)	\(=\)	\(0,\)	\(2y\)	\(=\)	\(-9xz^2 - 15z^3\)	\(0.056211\)	\(\infty\)

2-torsion field: 6.2.15990504192.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(1\)
Mordell-Weil rank:	\(1\)
2-Selmer rank:	\(1\)
Regulator:	\( 0.056211 \)
Real period:	\( 7.243718 \)
Tamagawa product:	\( 2 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 0.814361 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(3\)	\(4\)	\(5\)	\(2\)	\(1 + T\)
\(13\)	\(2\)	\(2\)	\(1\)	\(1 + 3 T + 13 T^{2}\)

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?
\(2\)	2.20.2	no

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 \(\Q\).