Genus 2 curve 86516.a.346064.1

• Label: 86516.a.346064.1
• Conductor: $86516$
• Discriminant: $-346064$
• 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

$y^2 + y = x^6 - 4x^5 + 2x^4 + 2x^3 - x$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(86516\)	\(=\)	\( 2^{2} \cdot 43 \cdot 503 \)
Discriminant:	\( \Delta \)	\(=\)	\(-346064\)	\(=\)	\( - 2^{4} \cdot 43 \cdot 503 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(248\)	\(=\)	\( 2^{3} \cdot 31 \)
\( I_4 \)	\(=\)	\(2773\)	\(=\)	\( 47 \cdot 59 \)
\( I_6 \)	\(=\)	\(170663\)	\(=\)	\( 17 \cdot 10039 \)
\( I_{10} \)	\(=\)	\(-43258\)	\(=\)	\( - 2 \cdot 43 \cdot 503 \)

Automorphism group

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

Rational points

Known points
\((1 : -1 : 0)\)	\((1 : 1 : 0)\)	\((0 : 0 : 1)\)	\((0 : -1 : 1)\)	\((1 : 0 : 1)\)	\((1 : -1 : 1)\)
\((-1 : 2 : 1)\)	\((-1 : -3 : 1)\)	\((-1 : 3 : 2)\)	\((1 : -3 : 2)\)	\((3 : -3 : 4)\)	\((1 : -5 : 2)\)
\((-1 : -11 : 2)\)	\((1 : -13 : 3)\)	\((-2 : 14 : 1)\)	\((1 : -14 : 3)\)	\((-2 : -15 : 1)\)	\((3 : -61 : 4)\)
\((5 : -3469 : 24)\)	\((5 : -10355 : 24)\)

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
\(2 \cdot(0 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\)	\(x^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(xz^2 - z^3\)	\(0.358125\)	\(\infty\)
\((0 : -1 : 1) + (1 : -3 : 2) - (1 : -1 : 0) - (1 : 1 : 0)\)	\(x (2x - z)\)	\(=\)	\(0,\)	\(4y\)	\(=\)	\(5xz^2 - 4z^3\)	\(0.276448\)	\(\infty\)
\((0 : -1 : 1) - (1 : 1 : 0)\)	\(z x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-x^3 - z^3\)	\(0.166828\)	\(\infty\)

2-torsion field: 6.4.5537024.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(3\)   (upper bound)
Mordell-Weil rank:	\(3\)
2-Selmer rank:	\(3\)
Regulator:	\( 0.015669 \)
Real period:	\( 17.79644 \)
Tamagawa product:	\( 3 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 0.836557 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(2\)	\(4\)	\(3\)	\(1 + 2 T + 2 T^{2}\)
\(43\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 - T + 43 T^{2} )\)
\(503\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 + 30 T + 503 T^{2} )\)

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 \(\Q\).