Genus 2 curve 82325.b.82325.1

• Label: 82325.b.82325.1
• Conductor: $82325$
• Discriminant: $82325$
• 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

$y^2 + (x^3 + x + 1)y = x^5 - 2x^4 + 3x^3 - 2x^2 + x$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(82325\)	\(=\)	\( 5^{2} \cdot 37 \cdot 89 \)
Discriminant:	\( \Delta \)	\(=\)	\(82325\)	\(=\)	\( 5^{2} \cdot 37 \cdot 89 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(612\)	\(=\)	\( 2^{2} \cdot 3^{2} \cdot 17 \)
\( I_4 \)	\(=\)	\(34305\)	\(=\)	\( 3 \cdot 5 \cdot 2287 \)
\( I_6 \)	\(=\)	\(8921025\)	\(=\)	\( 3^{2} \cdot 5^{2} \cdot 31 \cdot 1279 \)
\( I_{10} \)	\(=\)	\(10537600\)	\(=\)	\( 2^{7} \cdot 5^{2} \cdot 37 \cdot 89 \)

Automorphism group

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

Rational points

Known points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1)\)

Number of rational Weierstrass points: \(0\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

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

Generator	$D_0$						Height	Order
\((0 : -1 : 1) - (1 : -1 : 0)\)	\(z x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-z^3\)	\(0.759945\)	\(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\)	\(x^2 - xz + z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0.137678\)	\(\infty\)

2-torsion field: 6.2.82325.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(2\)
Mordell-Weil rank:	\(2\)
2-Selmer rank:	\(2\)
Regulator:	\( 0.102439 \)
Real period:	\( 9.079435 \)
Tamagawa product:	\( 1 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 0.930091 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(5\)	\(2\)	\(2\)	\(1\)	\(1 + T + 5 T^{2}\)
\(37\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 - 2 T + 37 T^{2} )\)
\(89\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 - 14 T + 89 T^{2} )\)

Galois representations

The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) .

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 \(\Q\).