Genus 2 curve 758059.a.758059.1

• Label: 758059.a.758059.1
• Conductor: $758059$
• Discriminant: $758059$
• 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

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

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(758059\)	\(=\)	\( 53 \cdot 14303 \)
Discriminant:	\( \Delta \)	\(=\)	\(758059\)	\(=\)	\( 53 \cdot 14303 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(604\)	\(=\)	\( 2^{2} \cdot 151 \)
\( I_4 \)	\(=\)	\(82729\)	\(=\)	\( 82729 \)
\( I_6 \)	\(=\)	\(30781947\)	\(=\)	\( 3 \cdot 7^{2} \cdot 209401 \)
\( I_{10} \)	\(=\)	\(-97031552\)	\(=\)	\( - 2^{7} \cdot 53 \cdot 14303 \)

Automorphism group

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

Rational points

Known points
\((1 : 0 : 0)\)	\((1 : -1 : 0)\)	\((-1 : 0 : 1)\)	\((0 : 1 : 1)\)	\((-1 : 1 : 1)\)	\((1 : -1 : 1)\)
\((0 : -2 : 1)\)	\((1 : 0 : 2)\)	\((2 : 0 : 1)\)	\((1 : -2 : 1)\)	\((4 : 1 : 1)\)	\((-2 : 4 : 1)\)
\((-2 : 5 : 1)\)	\((5 : -8 : 3)\)	\((2 : -11 : 1)\)	\((1 : -13 : 2)\)	\((4 : -70 : 1)\)	\((-1 : 81 : 4)\)
\((-1 : -128 : 4)\)	\((5 : -189 : 3)\)	\((5 : -224 : 6)\)	\((5 : -297 : 6)\)	\((16 : -7722 : 21)\)	\((16 : -12691 : 21)\)

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
\((0 : -2 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\)	\(x (x - z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(xz^2 - 2z^3\)	\(0.788653\)	\(\infty\)
\((1 : -2 : 1) - (1 : 0 : 0)\)	\(z (x - z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-x^3 - z^3\)	\(0.368877\)	\(\infty\)
\((-1 : 0 : 1) + (1 : -2 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\)	\((x - z) (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-xz^2 - z^3\)	\(0.653970\)	\(\infty\)
\((1 : -2 : 1) - (1 : -1 : 0)\)	\(z (x - z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-2z^3\)	\(0.641855\)	\(\infty\)

2-torsion field: 6.2.48515776.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(4\)   (upper bound)
Mordell-Weil rank:	\(4\)
2-Selmer rank:	\(4\)
Regulator:	\( 0.094569 \)
Real period:	\( 15.68563 \)
Tamagawa product:	\( 1 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 1.483376 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(53\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 + 10 T + 53 T^{2} )\)
\(14303\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 - 150 T + 14303 T^{2} )\)

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 \(\Q\).