Genus 2 curve 51035.a.255175.1

• Label: 51035.a.255175.1
• Conductor: $51035$
• Discriminant: $-255175$
• Mordell-Weil group: \(\Z \oplus \Z \oplus \Z\)
• 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

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

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(51035\)	\(=\)	\( 5 \cdot 59 \cdot 173 \)
Discriminant:	\( \Delta \)	\(=\)	\(-255175\)	\(=\)	\( - 5^{2} \cdot 59 \cdot 173 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(1572\)	\(=\)	\( 2^{2} \cdot 3 \cdot 131 \)
\( I_4 \)	\(=\)	\(70521\)	\(=\)	\( 3 \cdot 11 \cdot 2137 \)
\( I_6 \)	\(=\)	\(31551645\)	\(=\)	\( 3 \cdot 5 \cdot 31 \cdot 67853 \)
\( I_{10} \)	\(=\)	\(-32662400\)	\(=\)	\( - 2^{7} \cdot 5^{2} \cdot 59 \cdot 173 \)

Automorphism group

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

Rational points

Known points
\((1 : 0 : 0)\)	\((1 : -1 : 0)\)	\((0 : 0 : 1)\)	\((-1 : 0 : 1)\)	\((0 : -1 : 1)\)	\((-1 : 1 : 1)\)
\((1 : 1 : 1)\)	\((-1 : 1 : 2)\)	\((2 : -3 : 1)\)	\((1 : -4 : 1)\)	\((-1 : -4 : 2)\)	\((2 : -8 : 1)\)
\((-3 : 12 : 1)\)	\((5 : -15 : 1)\)	\((-3 : 17 : 1)\)	\((2 : 27 : 3)\)	\((2 : -80 : 3)\)	\((5 : -116 : 1)\)
\((7 : 17072 : 36)\)	\((7 : -73143 : 36)\)

Number of rational Weierstrass points: \(0\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

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

Generator	$D_0$						Height	Order
\((0 : 0 : 1) + (2 : -8 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\)	\(x (x - 2z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-4xz^2\)	\(0.507487\)	\(\infty\)
\((-1 : 1 : 2) - (1 : 0 : 0)\)	\(z (2x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-x^3\)	\(0.222465\)	\(\infty\)
\((-1 : 0 : 1) + (2 : -3 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\)	\((x - 2z) (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-xz^2 - z^3\)	\(0.159942\)	\(\infty\)

2-torsion field: 6.4.653248.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(3\)   (upper bound)
Mordell-Weil rank:	\(3\)
2-Selmer rank:	\(3\)
Regulator:	\( 0.014614 \)
Real period:	\( 18.56390 \)
Tamagawa product:	\( 2 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 0.542622 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(5\)	\(1\)	\(2\)	\(2\)	\(( 1 + T )( 1 + T + 5 T^{2} )\)
\(59\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 + 14 T + 59 T^{2} )\)
\(173\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 + 2 T + 173 T^{2} )\)

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 \(\Q\).