Genus 2 curve 24320.b.243200.1

• Label: 24320.b.243200.1
• Conductor: $24320$
• Discriminant: $243200$
• Mordell-Weil group: \(\Z \oplus \Z \oplus \Z/{2}\Z\)
• 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

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

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(24320\)	\(=\)	\( 2^{8} \cdot 5 \cdot 19 \)
Discriminant:	\( \Delta \)	\(=\)	\(243200\)	\(=\)	\( 2^{9} \cdot 5^{2} \cdot 19 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(6\)	\(=\)	\( 2 \cdot 3 \)
\( I_4 \)	\(=\)	\(54\)	\(=\)	\( 2 \cdot 3^{3} \)
\( I_6 \)	\(=\)	\(-792\)	\(=\)	\( - 2^{3} \cdot 3^{2} \cdot 11 \)
\( I_{10} \)	\(=\)	\(950\)	\(=\)	\( 2 \cdot 5^{2} \cdot 19 \)

Automorphism group

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

Rational points

Known points
\((1 : 0 : 0)\)	\((0 : 0 : 1)\)	\((-1 : 0 : 1)\)	\((0 : -1 : 1)\)	\((-1 : -1 : 1)\)	\((-1 : 0 : 2)\)
\((1 : 2 : 1)\)	\((1 : -3 : 1)\)	\((-1 : -8 : 2)\)	\((3 : 27 : 1)\)	\((3 : -28 : 1)\)	\((20 : 2624 : 1)\)
\((20 : -2625 : 1)\)

Number of rational Weierstrass points: \(1\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

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

Generator	$D_0$						Height	Order
\((-1 : -1 : 1) + (0 : -1 : 1) - 2 \cdot(1 : 0 : 0)\)	\(x (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-z^3\)	\(0.421492\)	\(\infty\)
\((0 : -1 : 1) - (1 : 0 : 0)\)	\(x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-z^3\)	\(0.121889\)	\(\infty\)
\(D_0 - 2 \cdot(1 : 0 : 0)\)	\(2x^2 + 2xz + z^2\)	\(=\)	\(0,\)	\(2y\)	\(=\)	\(-z^3\)	\(0\)	\(2\)

2-torsion field: 6.0.369664.2

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(2\)
Mordell-Weil rank:	\(2\)
2-Selmer rank:	\(3\)
Regulator:	\( 0.047868 \)
Real period:	\( 12.98001 \)
Tamagawa product:	\( 4 \)
Torsion order:	\( 2 \)
Leading coefficient:	\( 0.621329 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(8\)	\(9\)	\(2\)	\(1 + 2 T + 2 T^{2}\)
\(5\)	\(1\)	\(2\)	\(2\)	\(( 1 - T )( 1 + 4 T + 5 T^{2} )\)
\(19\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 + 6 T + 19 T^{2} )\)

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?
\(2\)	2.60.1	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 \(\Q\).