Genus 2 curve 6982.a.13964.1

• Label: 6982.a.13964.1
• Conductor: $6982$
• Discriminant: $-13964$
• 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

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

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(6982\)	\(=\)	\( 2 \cdot 3491 \)
Discriminant:	\( \Delta \)	\(=\)	\(-13964\)	\(=\)	\( - 2^{2} \cdot 3491 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(820\)	\(=\)	\( 2^{2} \cdot 5 \cdot 41 \)
\( I_4 \)	\(=\)	\(12505\)	\(=\)	\( 5 \cdot 41 \cdot 61 \)
\( I_6 \)	\(=\)	\(2944285\)	\(=\)	\( 5 \cdot 263 \cdot 2239 \)
\( I_{10} \)	\(=\)	\(-1787392\)	\(=\)	\( - 2^{9} \cdot 3491 \)

Automorphism group

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

Rational points

Known points
\((1 : 0 : 0)\)	\((1 : -1 : 0)\)	\((0 : 0 : 1)\)	\((-1 : 0 : 1)\)	\((0 : -1 : 1)\)	\((1 : 0 : 1)\)
\((1 : -2 : 1)\)	\((-2 : 3 : 1)\)	\((-2 : 4 : 1)\)	\((-3 : 12 : 1)\)	\((-3 : 14 : 1)\)	\((13 : 408 : 5)\)
\((13 : -2730 : 5)\)

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
\((1 : -2 : 1) - (1 : 0 : 0)\)	\(z (x - z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-x^3 - z^3\)	\(0.474383\)	\(\infty\)
\((-1 : 0 : 1) - (1 : 0 : 0)\)	\(z (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-x^3 - z^3\)	\(0.015759\)	\(\infty\)

2-torsion field: 5.3.55856.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(2\)
Mordell-Weil rank:	\(2\)
2-Selmer rank:	\(2\)
Regulator:	\( 0.007467 \)
Real period:	\( 23.52533 \)
Tamagawa product:	\( 2 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 0.351346 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(1\)	\(2\)	\(2\)	\(( 1 + T )( 1 + T + 2 T^{2} )\)
\(3491\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 + 92 T + 3491 T^{2} )\)

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 \(\Q\).