Genus 2 curve 100130.a.200260.1

• Label: 100130.a.200260.1
• Conductor: $100130$
• Discriminant: $-200260$
• 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

$y^2 + (x^2 + x + 1)y = x^6 + 5x^5 + 6x^4 + x^2 + x$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(100130\)	\(=\)	\( 2 \cdot 5 \cdot 17 \cdot 19 \cdot 31 \)
Discriminant:	\( \Delta \)	\(=\)	\(-200260\)	\(=\)	\( - 2^{2} \cdot 5 \cdot 17 \cdot 19 \cdot 31 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(532\)	\(=\)	\( 2^{2} \cdot 7 \cdot 19 \)
\( I_4 \)	\(=\)	\(218185\)	\(=\)	\( 5 \cdot 11 \cdot 3967 \)
\( I_6 \)	\(=\)	\(-1085371\)	\(=\)	\( - 7 \cdot 47 \cdot 3299 \)
\( I_{10} \)	\(=\)	\(-25633280\)	\(=\)	\( - 2^{9} \cdot 5 \cdot 17 \cdot 19 \cdot 31 \)

Automorphism group

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

Rational points

Known points
\((1 : -1 : 0)\)	\((1 : 1 : 0)\)	\((0 : 0 : 1)\)	\((0 : -1 : 1)\)	\((-1 : 1 : 1)\)	\((-1 : -2 : 1)\)
\((7 : 602 : 3)\)	\((7 : -839 : 3)\)

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
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\)	\(2x^2 + 5xz + z^2\)	\(=\)	\(0,\)	\(4y\)	\(=\)	\(5xz^2 - z^3\)	\(0.429826\)	\(\infty\)
\((-1 : -2 : 1) - (1 : 1 : 0)\)	\(z (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-x^3 - 3z^3\)	\(0.300725\)	\(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\)	\(x^2 + 3xz + z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(xz^2\)	\(0\)	\(2\)

2-torsion field: 6.2.200520338000.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(2\)
Mordell-Weil rank:	\(2\)
2-Selmer rank:	\(3\)
Regulator:	\( 0.128156 \)
Real period:	\( 18.88632 \)
Tamagawa product:	\( 2 \)
Torsion order:	\( 2 \)
Leading coefficient:	\( 1.210198 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(1\)	\(2\)	\(2\)	\(( 1 + T )( 1 + T + 2 T^{2} )\)
\(5\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 + 5 T^{2} )\)
\(17\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 + 6 T + 17 T^{2} )\)
\(19\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 + 6 T + 19 T^{2} )\)
\(31\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 + 6 T + 31 T^{2} )\)

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?
\(2\)	2.15.1	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 \(\Q\).