Genus 2 curve 18080.c.723200.1

• Label: 18080.c.723200.1
• Conductor: $18080$
• Discriminant: $-723200$
• 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

$y^2 + xy = x^6 - 2x^5 + x^3 + x^2 - 2x + 1$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(18080\)	\(=\)	\( 2^{5} \cdot 5 \cdot 113 \)
Discriminant:	\( \Delta \)	\(=\)	\(-723200\)	\(=\)	\( - 2^{8} \cdot 5^{2} \cdot 113 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(148\)	\(=\)	\( 2^{2} \cdot 37 \)
\( I_4 \)	\(=\)	\(2917\)	\(=\)	\( 2917 \)
\( I_6 \)	\(=\)	\(148783\)	\(=\)	\( 148783 \)
\( I_{10} \)	\(=\)	\(90400\)	\(=\)	\( 2^{5} \cdot 5^{2} \cdot 113 \)

Automorphism group

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

Rational points

Known points
\((1 : -1 : 0)\)	\((1 : 1 : 0)\)	\((0 : -1 : 1)\)	\((0 : 1 : 1)\)	\((1 : 0 : 1)\)	\((1 : -1 : 1)\)
\((-1 : -2 : 1)\)	\((-1 : 3 : 1)\)	\((1 : 3 : 2)\)	\((3 : -1 : 2)\)	\((1 : -7 : 2)\)	\((3 : -11 : 2)\)
\((7 : -41 : 6)\)	\((7 : -211 : 6)\)

Number of rational Weierstrass points: \(0\)

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
\((0 : -1 : 1) - (1 : 1 : 0)\)	\(z x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-x^3 - z^3\)	\(0.557085\)	\(\infty\)
\((0 : -1 : 1) + (3 : -11 : 2) - (1 : -1 : 0) - (1 : 1 : 0)\)	\(x (2x - 3z)\)	\(=\)	\(0,\)	\(4y\)	\(=\)	\(-xz^2 - 4z^3\)	\(0.039896\)	\(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\)	\(2x^2 - 2xz + z^2\)	\(=\)	\(0,\)	\(2y\)	\(=\)	\(-xz^2\)	\(0\)	\(2\)

2-torsion field: 4.0.1808.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(2\)
Mordell-Weil rank:	\(2\)
2-Selmer rank:	\(3\)
Regulator:	\( 0.022196 \)
Real period:	\( 14.76743 \)
Tamagawa product:	\( 8 \)
Torsion order:	\( 2 \)
Leading coefficient:	\( 0.655575 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(5\)	\(8\)	\(4\)	\(1 + T\)
\(5\)	\(1\)	\(2\)	\(2\)	\(( 1 + T )( 1 + 2 T + 5 T^{2} )\)
\(113\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 + 14 T + 113 T^{2} )\)

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?
\(2\)	2.90.1	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 \(\Q\).