Genus 2 curve 12961.a.168493.1

• Label: 12961.a.168493.1
• Conductor: $12961$
• Discriminant: $168493$
• 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

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

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(12961\)	\(=\)	\( 13 \cdot 997 \)
Discriminant:	\( \Delta \)	\(=\)	\(168493\)	\(=\)	\( 13^{2} \cdot 997 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(760\)	\(=\)	\( 2^{3} \cdot 5 \cdot 19 \)
\( I_4 \)	\(=\)	\(37744\)	\(=\)	\( 2^{4} \cdot 7 \cdot 337 \)
\( I_6 \)	\(=\)	\(7078728\)	\(=\)	\( 2^{3} \cdot 3 \cdot 294947 \)
\( I_{10} \)	\(=\)	\(673972\)	\(=\)	\( 2^{2} \cdot 13^{2} \cdot 997 \)

Automorphism group

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

Rational points

Known points
\((1 : 0 : 0)\)	\((0 : 0 : 1)\)	\((0 : -1 : 1)\)	\((1 : 0 : 1)\)	\((1 : -1 : 1)\)	\((-1 : 2 : 1)\)
\((-1 : -3 : 1)\)	\((2 : 6 : 1)\)	\((2 : -7 : 1)\)	\((-3 : 11 : 1)\)	\((-3 : -12 : 1)\)

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
\((0 : 0 : 1) + (2 : 6 : 1) - 2 \cdot(1 : 0 : 0)\)	\(x (x - 2z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(3xz^2\)	\(0.262999\)	\(\infty\)
\((1 : 0 : 1) + (2 : 6 : 1) - 2 \cdot(1 : 0 : 0)\)	\((x - 2z) (x - z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(6xz^2 - 6z^3\)	\(0.043869\)	\(\infty\)

2-torsion field: 5.1.15952.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(2\)
Mordell-Weil rank:	\(2\)
2-Selmer rank:	\(2\)
Regulator:	\( 0.011482 \)
Real period:	\( 21.24043 \)
Tamagawa product:	\( 2 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 0.487783 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(13\)	\(1\)	\(2\)	\(2\)	\(( 1 + T )( 1 + 6 T + 13 T^{2} )\)
\(997\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 - 8 T + 997 T^{2} )\)

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 \(\Q\).