Genus 2 curve 816.a.13872.1

• Label: 816.a.13872.1
• Conductor: $816$
• Discriminant: $-13872$
• Mordell-Weil group: \(\Z/{2}\Z \oplus \Z/{6}\Z\)
• 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

$y^2 + (x^3 + x^2)y = -2x^4 + 6x^2 - 8x + 3$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(816\)	\(=\)	\( 2^{4} \cdot 3 \cdot 17 \)
Discriminant:	\( \Delta \)	\(=\)	\(-13872\)	\(=\)	\( - 2^{4} \cdot 3 \cdot 17^{2} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(688\)	\(=\)	\( 2^{4} \cdot 43 \)
\( I_4 \)	\(=\)	\(9592\)	\(=\)	\( 2^{3} \cdot 11 \cdot 109 \)
\( I_6 \)	\(=\)	\(2944404\)	\(=\)	\( 2^{2} \cdot 3^{3} \cdot 137 \cdot 199 \)
\( I_{10} \)	\(=\)	\(55488\)	\(=\)	\( 2^{6} \cdot 3 \cdot 17^{2} \)

Automorphism group

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

Rational points

All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (1 : -1 : 1),\, (-3 : 9 : 1)\)

Number of rational Weierstrass points: \(2\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z/{2}\Z \oplus \Z/{6}\Z\)

Generator	$D_0$						Height	Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\)	\(x^2 + 2xz - 2z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-2xz^2 + z^3\)	\(0\)	\(2\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\)	\(2x^2 - z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(2xz^2 - 2z^3\)	\(0\)	\(6\)

2-torsion field: \(\Q(\zeta_{12})\)

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(2\)
Regulator:	\( 1 \)
Real period:	\( 22.16669 \)
Tamagawa product:	\( 2 \)
Torsion order:	\( 12 \)
Leading coefficient:	\( 0.307870 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(4\)	\(4\)	\(1\)	\(1 + T\)
\(3\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 + 2 T + 3 T^{2} )\)
\(17\)	\(1\)	\(2\)	\(2\)	\(( 1 - T )( 1 + 6 T + 17 T^{2} )\)

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?
\(2\)	2.180.3	yes
\(3\)	3.80.1	yes

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 \(\Q\).