Genus 2 curve 25570.b.255700.1

• Label: 25570.b.255700.1
• Conductor: $25570$
• Discriminant: $255700$
• 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

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

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(25570\)	\(=\)	\( 2 \cdot 5 \cdot 2557 \)
Discriminant:	\( \Delta \)	\(=\)	\(255700\)	\(=\)	\( 2^{2} \cdot 5^{2} \cdot 2557 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(4084\)	\(=\)	\( 2^{2} \cdot 1021 \)
\( I_4 \)	\(=\)	\(60001\)	\(=\)	\( 29 \cdot 2069 \)
\( I_6 \)	\(=\)	\(62743873\)	\(=\)	\( 62743873 \)
\( I_{10} \)	\(=\)	\(32729600\)	\(=\)	\( 2^{9} \cdot 5^{2} \cdot 2557 \)

Automorphism group

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

Rational points

Known points
\((1 : 0 : 0)\)	\((1 : -1 : 0)\)	\((-1 : 0 : 1)\)	\((1 : 0 : 1)\)	\((1 : -2 : 1)\)	\((3 : -5 : 2)\)
\((-3 : 12 : 1)\)	\((-3 : 14 : 1)\)	\((-5 : -24 : 3)\)	\((3 : -30 : 2)\)	\((-11 : -75 : 6)\)	\((-5 : 122 : 3)\)
\((-11 : 1190 : 6)\)

Number of rational Weierstrass points: \(1\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

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

Generator	$D_0$						Height	Order
\((1 : -2 : 1) - (1 : -1 : 0)\)	\(z (x - z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-2z^3\)	\(0.448852\)	\(\infty\)
\((-1 : 0 : 1) - (1 : 0 : 0)\)	\(z (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-x^3 - z^3\)	\(0.034669\)	\(\infty\)

2-torsion field: 5.1.40912.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(2\)
Mordell-Weil rank:	\(2\)
2-Selmer rank:	\(2\)
Regulator:	\( 0.015541 \)
Real period:	\( 15.87321 \)
Tamagawa product:	\( 4 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 0.986763 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(1\)	\(2\)	\(2\)	\(( 1 + T )( 1 - T + 2 T^{2} )\)
\(5\)	\(1\)	\(2\)	\(2\)	\(( 1 - T )( 1 + 4 T + 5 T^{2} )\)
\(2557\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 + 7 T + 2557 T^{2} )\)

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?
\(2\)	2.6.1	no

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 \(\Q\).