Genus 2 Curve 15256.a.122048.1

• Label: 15256.a.122048.1
• Conductor: 15256
• Discriminant: 122048
• Mordell-Weil group: \(\Z \times \Z\)
• Sato-Tate group: $\mathrm{USp}(4)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\R\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\Q\)
• \(\overline{\Q}\)-simple: yes
• \(\mathrm{GL}_2\)-type: no

Minimal equation

$y^2 + (x + 1)y = x^6 - x^4 + 2x^2 + x$

(, )

Invariants

Conductor:	\( N \)	=	\(15256\)	=	\( 2^{3} \cdot 1907 \)
Discriminant:	\( \Delta \)	=	\(122048\)	=	\( 2^{6} \cdot 1907 \)

Igusa-Clebsch invariants

\( I_2 \)	=	\(-384\)	=	\( - 2^{7} \cdot 3 \)
\( I_4 \)	=	\(67776\)	=	\( 2^{6} \cdot 3 \cdot 353 \)
\( I_6 \)	=	\(-13663488\)	=	\( - 2^{8} \cdot 3 \cdot 17791 \)
\( I_{10} \)	=	\(499908608\)	=	\( 2^{18} \cdot 1907 \)

Automorphism group

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

Rational points

Known points
\((1 : -1 : 0)\)	\((1 : 1 : 0)\)	\((0 : 0 : 1)\)	\((0 : -1 : 1)\)	\((-1 : -1 : 1)\)	\((-1 : 1 : 1)\)
\((1 : 1 : 1)\)	\((-1 : -1 : 2)\)	\((1 : -3 : 1)\)	\((-1 : -3 : 2)\)	\((-1 : -17 : 4)\)	\((2 : 17 : 3)\)
\((-1 : -31 : 4)\)	\((2 : -62 : 3)\)

Number of rational Weierstrass points: \(0\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian:

Group structure: \(\Z \times \Z\)

Generator	$D_0$						Height	Order
\(2 \cdot(0 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\)	\(x^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(xz^2\)	\(0.264086\)	\(\infty\)
\((-1 : -1 : 1) - (1 : -1 : 0)\)	\(z (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(x^3\)	\(0.021732\)	\(\infty\)

2-torsion field: 6.2.488192.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(2\)
Mordell-Weil rank:	\(2\)
2-Selmer rank:	\(2\)
Regulator:	\( 0.005663 \)
Real period:	\( 17.79322 \)
Tamagawa product:	\( 6 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 0.604627 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor
\(2\)	\(6\)	\(3\)	\(6\)	\(1 + T\)
\(1907\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 - 29 T + 1907 T^{2} )\)

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 \(\Q\).