Genus 2 curve 18080.b.180800.1

• Label: 18080.b.180800.1
• Conductor: $18080$
• Discriminant: $-180800$
• Mordell-Weil group: \(\Z \oplus \Z/{2}\Z\)
• 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

$y^2 + (x^3 + x)y = -x^4 - x^3 + x^2 - x + 2$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(18080\)	\(=\)	\( 2^{5} \cdot 5 \cdot 113 \)
Discriminant:	\( \Delta \)	\(=\)	\(-180800\)	\(=\)	\( - 2^{6} \cdot 5^{2} \cdot 113 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(208\)	\(=\)	\( 2^{4} \cdot 13 \)
\( I_4 \)	\(=\)	\(2563\)	\(=\)	\( 11 \cdot 233 \)
\( I_6 \)	\(=\)	\(152516\)	\(=\)	\( 2^{2} \cdot 7 \cdot 13 \cdot 419 \)
\( I_{10} \)	\(=\)	\(22600\)	\(=\)	\( 2^{3} \cdot 5^{2} \cdot 113 \)

Automorphism group

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

Rational points

All points
\((1 : 0 : 0)\)	\((1 : -1 : 0)\)	\((1 : 0 : 1)\)	\((-2 : 0 : 1)\)	\((1 : -2 : 1)\)	\((-2 : 10 : 1)\)
\((89 : 42589520 : 329)\)	\((89 : -52927938 : 329)\)

Number of rational Weierstrass points: \(0\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

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

Generator	$D_0$						Height	Order
\((1 : 0 : 1) - (1 : 0 : 0)\)	\(z (x - z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-x^3 + z^3\)	\(0.061051\)	\(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\)	\(x^2 + z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0\)	\(2\)

2-torsion field: 4.0.1808.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(1\)
Mordell-Weil rank:	\(1\)
2-Selmer rank:	\(2\)
Regulator:	\( 0.061051 \)
Real period:	\( 15.52813 \)
Tamagawa product:	\( 4 \)
Torsion order:	\( 2 \)
Leading coefficient:	\( 0.948017 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(5\)	\(6\)	\(2\)	\(1 + T + 2 T^{2}\)
\(5\)	\(1\)	\(2\)	\(2\)	\(( 1 - T )( 1 + 2 T + 5 T^{2} )\)
\(113\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 - 2 T + 113 T^{2} )\)

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 \(\Q\).