Genus 2 curve 131074.a.262148.1

• Label: 131074.a.262148.1
• Conductor: $131074$
• Discriminant: $-262148$
• 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

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

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(131074\)	\(=\)	\( 2 \cdot 65537 \)
Discriminant:	\( \Delta \)	\(=\)	\(-262148\)	\(=\)	\( - 2^{2} \cdot 65537 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(940\)	\(=\)	\( 2^{2} \cdot 5 \cdot 47 \)
\( I_4 \)	\(=\)	\(32425\)	\(=\)	\( 5^{2} \cdot 1297 \)
\( I_6 \)	\(=\)	\(9977067\)	\(=\)	\( 3^{3} \cdot 163 \cdot 2267 \)
\( I_{10} \)	\(=\)	\(33554944\)	\(=\)	\( 2^{9} \cdot 65537 \)

Automorphism group

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

Rational points

Known points
\((1 : 0 : 0)\)	\((1 : -1 : 0)\)	\((0 : 1 : 1)\)	\((1 : 0 : 1)\)	\((0 : -2 : 1)\)	\((-2 : 1 : 1)\)
\((-1 : -2 : 1)\)	\((-1 : 2 : 1)\)	\((1 : -2 : 1)\)	\((2 : -2 : 1)\)	\((-2 : 6 : 1)\)	\((2 : -7 : 1)\)
\((5 : 63 : 6)\)	\((7 : -64 : 3)\)	\((7 : -306 : 3)\)	\((5 : -404 : 6)\)	\((43 : -4332 : 10)\)	\((43 : -76175 : 10)\)

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
\((-1 : -2 : 1) - (1 : 0 : 0)\)	\(z (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-x^3 - 3z^3\)	\(0.597671\)	\(\infty\)
\((1 : 0 : 1) - (1 : 0 : 0)\)	\(z (x - z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-x^3 + z^3\)	\(0.326635\)	\(\infty\)
\(D_0 - 2 \cdot(1 : -1 : 0)\)	\(z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0.207687\)	\(\infty\)

2-torsion field: 6.0.4194368.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(3\)   (upper bound)
Mordell-Weil rank:	\(3\)
2-Selmer rank:	\(3\)
Regulator:	\( 0.032962 \)
Real period:	\( 15.75149 \)
Tamagawa product:	\( 2 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 1.038422 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(1\)	\(2\)	\(2\)	\(( 1 + T )( 1 + T + 2 T^{2} )\)
\(65537\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 - 295 T + 65537 T^{2} )\)

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 \(\Q\).