Genus 2 curve 150456.a.902736.1

• Label: 150456.a.902736.1
• Conductor: $150456$
• Discriminant: $-902736$
• 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

$y^2 + y = x^6 - 2x^5 + x$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(150456\)	\(=\)	\( 2^{3} \cdot 3 \cdot 6269 \)
Discriminant:	\( \Delta \)	\(=\)	\(-902736\)	\(=\)	\( - 2^{4} \cdot 3^{2} \cdot 6269 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(280\)	\(=\)	\( 2^{3} \cdot 5 \cdot 7 \)
\( I_4 \)	\(=\)	\(205\)	\(=\)	\( 5 \cdot 41 \)
\( I_6 \)	\(=\)	\(72365\)	\(=\)	\( 5 \cdot 41 \cdot 353 \)
\( I_{10} \)	\(=\)	\(112842\)	\(=\)	\( 2 \cdot 3^{2} \cdot 6269 \)

Automorphism group

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

Rational points

Known points
\((1 : -1 : 0)\)	\((1 : 1 : 0)\)	\((0 : 0 : 1)\)	\((0 : -1 : 1)\)	\((1 : 0 : 1)\)	\((-1 : 1 : 1)\)
\((1 : -1 : 1)\)	\((-1 : -2 : 1)\)	\((2 : 1 : 1)\)	\((2 : -2 : 1)\)	\((1 : 7 : 3)\)	\((-3 : -7 : 4)\)
\((-1 : -29 : 4)\)	\((1 : -34 : 3)\)	\((-1 : -35 : 4)\)	\((2 : 37 : 5)\)	\((-3 : -57 : 4)\)	\((2 : -162 : 5)\)
\((-4 : -5069 : 33)\)	\((-4 : -30868 : 33)\)

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
\((0 : 0 : 1) - (1 : -1 : 0)\)	\(z x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(x^3\)	\(0.417859\)	\(\infty\)
\((0 : -1 : 1) - (1 : -1 : 0)\)	\(z x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(x^3 - z^3\)	\(0.364706\)	\(\infty\)
\((-1 : -2 : 1) - (1 : -1 : 0)\)	\(z (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(x^3 - z^3\)	\(0.157102\)	\(\infty\)

2-torsion field: 6.4.1604864.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(3\)   (upper bound)
Mordell-Weil rank:	\(3\)
2-Selmer rank:	\(3\)
Regulator:	\( 0.021395 \)
Real period:	\( 13.30252 \)
Tamagawa product:	\( 4 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 1.138450 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(3\)	\(4\)	\(2\)	\(1 + 2 T + 2 T^{2}\)
\(3\)	\(1\)	\(2\)	\(2\)	\(( 1 + T )( 1 + 2 T + 3 T^{2} )\)
\(6269\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 - 10 T + 6269 T^{2} )\)

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 \(\Q\).