Genus 2 curve 5331.a.15993.1

• Label: 5331.a.15993.1
• Conductor: $5331$
• Discriminant: $15993$
• 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

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

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(5331\)	\(=\)	\( 3 \cdot 1777 \)
Discriminant:	\( \Delta \)	\(=\)	\(15993\)	\(=\)	\( 3^{2} \cdot 1777 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(68\)	\(=\)	\( 2^{2} \cdot 17 \)
\( I_4 \)	\(=\)	\(8329\)	\(=\)	\( 8329 \)
\( I_6 \)	\(=\)	\(84469\)	\(=\)	\( 7 \cdot 11 \cdot 1097 \)
\( I_{10} \)	\(=\)	\(2047104\)	\(=\)	\( 2^{7} \cdot 3^{2} \cdot 1777 \)

Automorphism group

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

Rational points

Known points
\((1 : 0 : 0)\)	\((1 : -1 : 0)\)	\((0 : 0 : 1)\)	\((-1 : 0 : 1)\)	\((0 : -1 : 1)\)	\((1 : 0 : 1)\)
\((-1 : 1 : 1)\)	\((1 : -1 : 2)\)	\((1 : -3 : 1)\)	\((-2 : 3 : 1)\)	\((-2 : 6 : 1)\)	\((1 : -12 : 2)\)
\((1 : -28 : 6)\)	\((1 : -225 : 6)\)

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
\(D_0 - 2 \cdot(1 : -1 : 0)\)	\(z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0.256212\)	\(\infty\)
\((-1 : 0 : 1) + (1 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\)	\((x - z) (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0.022964\)	\(\infty\)

2-torsion field: 6.2.113728.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(2\)
Mordell-Weil rank:	\(2\)
2-Selmer rank:	\(2\)
Regulator:	\( 0.005863 \)
Real period:	\( 21.88764 \)
Tamagawa product:	\( 2 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 0.256679 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(3\)	\(1\)	\(2\)	\(2\)	\(( 1 + T )( 1 + 2 T + 3 T^{2} )\)
\(1777\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 - 67 T + 1777 T^{2} )\)

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 \(\Q\).