Genus 2 curve 86365.a.431825.1

• Label: 86365.a.431825.1
• Conductor: $86365$
• Discriminant: $431825$
• Mordell-Weil group: \(\Z \oplus \Z \oplus \Z\)
• 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

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

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(86365\)	\(=\)	\( 5 \cdot 23 \cdot 751 \)
Discriminant:	\( \Delta \)	\(=\)	\(431825\)	\(=\)	\( 5^{2} \cdot 23 \cdot 751 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(252\)	\(=\)	\( 2^{2} \cdot 3^{2} \cdot 7 \)
\( I_4 \)	\(=\)	\(65097\)	\(=\)	\( 3^{3} \cdot 2411 \)
\( I_6 \)	\(=\)	\(7313067\)	\(=\)	\( 3^{2} \cdot 73 \cdot 11131 \)
\( I_{10} \)	\(=\)	\(-55273600\)	\(=\)	\( - 2^{7} \cdot 5^{2} \cdot 23 \cdot 751 \)

Automorphism group

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

Rational points

Known points
\((1 : 0 : 0)\)	\((1 : -1 : 0)\)	\((-1 : 0 : 1)\)	\((0 : 1 : 1)\)	\((-1 : 1 : 1)\)	\((1 : 1 : 1)\)
\((0 : -2 : 1)\)	\((-2 : 2 : 1)\)	\((1 : -4 : 1)\)	\((1 : 6 : 2)\)	\((-2 : 7 : 1)\)	\((1 : -19 : 2)\)
\((5 : 28 : 1)\)	\((-4 : 34 : 3)\)	\((-4 : 39 : 3)\)	\((5 : -159 : 1)\)	\((-7 : 177 : 6)\)	\((-7 : 202 : 6)\)

Number of rational Weierstrass points: \(0\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

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

Generator	$D_0$						Height	Order
\((-1 : 0 : 1) + (0 : -2 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\)	\(x (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-2xz^2 - 2z^3\)	\(0.597586\)	\(\infty\)
\((-2 : 2 : 1) - (1 : -1 : 0)\)	\(z (x + 2z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(2z^3\)	\(0.208533\)	\(\infty\)
\((-2 : 2 : 1) + (-1 : 1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\)	\((x + z) (x + 2z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-xz^2\)	\(0.179162\)	\(\infty\)

2-torsion field: 6.2.5153512325417.3

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(3\)   (upper bound)
Mordell-Weil rank:	\(3\)
2-Selmer rank:	\(3\)
Regulator:	\( 0.022023 \)
Real period:	\( 16.31186 \)
Tamagawa product:	\( 2 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 0.718477 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(5\)	\(1\)	\(2\)	\(2\)	\(( 1 + T )( 1 + 2 T + 5 T^{2} )\)
\(23\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 + 9 T + 23 T^{2} )\)
\(751\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 + 6 T + 751 T^{2} )\)

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?
\(2\)	2.20.2	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 \(\Q\).