Genus 2 curve 792079.a.792079.1

• Label: 792079.a.792079.1
• Conductor: $792079$
• Discriminant: $-792079$
• Mordell-Weil group: \(\Z \oplus \Z \oplus \Z \oplus \Z\)
• 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

$y^2 + (x^3 + x + 1)y = -3x^5 + 7x^4 - 4x^2$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(792079\)	\(=\)	\( 41 \cdot 19319 \)
Discriminant:	\( \Delta \)	\(=\)	\(-792079\)	\(=\)	\( - 41 \cdot 19319 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(3012\)	\(=\)	\( 2^{2} \cdot 3 \cdot 251 \)
\( I_4 \)	\(=\)	\(432585\)	\(=\)	\( 3^{2} \cdot 5 \cdot 9613 \)
\( I_6 \)	\(=\)	\(340579413\)	\(=\)	\( 3^{2} \cdot 37 \cdot 1022761 \)
\( I_{10} \)	\(=\)	\(-101386112\)	\(=\)	\( - 2^{7} \cdot 41 \cdot 19319 \)

Automorphism group

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

Rational points

Known points
\((1 : 0 : 0)\)	\((1 : -1 : 0)\)	\((0 : 0 : 1)\)	\((0 : -1 : 1)\)	\((1 : 0 : 1)\)	\((2 : 0 : 1)\)
\((-1 : -2 : 1)\)	\((-2 : 0 : 3)\)	\((-1 : 3 : 1)\)	\((1 : -3 : 1)\)	\((-2 : -1 : 3)\)	\((1 : -6 : 2)\)
\((1 : -7 : 2)\)	\((3 : -9 : 1)\)	\((1 : -10 : 3)\)	\((2 : -11 : 1)\)	\((3 : -22 : 1)\)	\((1 : -27 : 3)\)
\((-1 : -39 : 8)\)	\((-1 : -408 : 8)\)	\((4 : -819 : 15)\)	\((4 : -3520 : 15)\)	\((13 : -4732 : 20)\)	\((-15 : -6579 : 14)\)
\((-15 : 10150 : 14)\)	\((13 : -10665 : 20)\)

Number of rational Weierstrass points: \(0\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z \oplus \Z \oplus \Z \oplus \Z\)

Generator	$D_0$						Height	Order
\((-1 : -2 : 1) + (1 : -7 : 2) - (1 : -1 : 0) - (1 : 0 : 0)\)	\((x + z) (2x - z)\)	\(=\)	\(0,\)	\(4y\)	\(=\)	\(3xz^2 - 5z^3\)	\(1.058260\)	\(\infty\)
\((0 : -1 : 1) - (1 : 0 : 0)\)	\(z x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-x^3 - z^3\)	\(0.382303\)	\(\infty\)
\((1 : -3 : 1) - (1 : -1 : 0)\)	\(z (x - z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-3z^3\)	\(0.767136\)	\(\infty\)
\((1 : -7 : 2) - (1 : 0 : 0)\)	\(z (2x - z)\)	\(=\)	\(0,\)	\(4y\)	\(=\)	\(-4x^3 - 3z^3\)	\(0.404632\)	\(\infty\)

2-torsion field: 6.4.50693056.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(4\)   (upper bound)
Mordell-Weil rank:	\(4\)
2-Selmer rank:	\(4\)
Regulator:	\( 0.084450 \)
Real period:	\( 18.00416 \)
Tamagawa product:	\( 1 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 1.520457 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(41\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 + 4 T + 41 T^{2} )\)
\(19319\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 + 40 T + 19319 T^{2} )\)

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 \(\Q\).