Genus 2 curve 15380.a.307600.1

• Label: 15380.a.307600.1
• Conductor: $15380$
• Discriminant: $307600$
• Mordell-Weil group: \(\Z \oplus \Z\)
• 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

$y^2 + xy = x^6 - 2x^5 + 2x^4 - 2x^3 + x$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(15380\)	\(=\)	\( 2^{2} \cdot 5 \cdot 769 \)
Discriminant:	\( \Delta \)	\(=\)	\(307600\)	\(=\)	\( 2^{4} \cdot 5^{2} \cdot 769 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(256\)	\(=\)	\( 2^{8} \)
\( I_4 \)	\(=\)	\(2764\)	\(=\)	\( 2^{2} \cdot 691 \)
\( I_6 \)	\(=\)	\(393028\)	\(=\)	\( 2^{2} \cdot 98257 \)
\( I_{10} \)	\(=\)	\(-1230400\)	\(=\)	\( - 2^{6} \cdot 5^{2} \cdot 769 \)

Automorphism group

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

Rational points

Known points
\((1 : -1 : 0)\)	\((1 : 1 : 0)\)	\((0 : 0 : 1)\)	\((1 : 0 : 1)\)	\((1 : -1 : 1)\)	\((-1 : -2 : 1)\)
\((-1 : 1 : 2)\)	\((-1 : 3 : 1)\)	\((-1 : 3 : 2)\)	\((1 : 3 : 2)\)	\((4 : 2 : 3)\)	\((1 : -7 : 2)\)
\((4 : -38 : 3)\)

Number of rational Weierstrass points: \(1\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z \oplus \Z\)

Generator	$D_0$						Height	Order
\((0 : 0 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\)	\(x (x - z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-xz^2\)	\(0.294277\)	\(\infty\)
\((1 : 0 : 1) + (1 : 3 : 2) - (1 : -1 : 0) - (1 : 1 : 0)\)	\((x - z) (2x - z)\)	\(=\)	\(0,\)	\(4y\)	\(=\)	\(-3xz^2 + 3z^3\)	\(0.023602\)	\(\infty\)

2-torsion field: 5.1.49216.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(2\)
Mordell-Weil rank:	\(2\)
2-Selmer rank:	\(2\)
Regulator:	\( 0.006830 \)
Real period:	\( 15.23203 \)
Tamagawa product:	\( 6 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 0.624298 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(2\)	\(4\)	\(3\)	\(1 + T + 2 T^{2}\)
\(5\)	\(1\)	\(2\)	\(2\)	\(( 1 - T )( 1 + 4 T + 5 T^{2} )\)
\(769\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 - 15 T + 769 T^{2} )\)

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?
\(2\)	2.6.1	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 \(\Q\).