Genus 2 curve 7945.c.198625.1

• Label: 7945.c.198625.1
• Conductor: $7945$
• Discriminant: $-198625$
• Mordell-Weil group: \(\Z \oplus \Z\)
• 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

$y^2 + (x^3 + x + 1)y = -5x^4 + 3x^3 + 7x^2 + 2x$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(7945\)	\(=\)	\( 5 \cdot 7 \cdot 227 \)
Discriminant:	\( \Delta \)	\(=\)	\(-198625\)	\(=\)	\( - 5^{3} \cdot 7 \cdot 227 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(4644\)	\(=\)	\( 2^{2} \cdot 3^{3} \cdot 43 \)
\( I_4 \)	\(=\)	\(123441\)	\(=\)	\( 3 \cdot 23 \cdot 1789 \)
\( I_6 \)	\(=\)	\(177145353\)	\(=\)	\( 3^{3} \cdot 7 \cdot 11 \cdot 139 \cdot 613 \)
\( I_{10} \)	\(=\)	\(-25424000\)	\(=\)	\( - 2^{7} \cdot 5^{3} \cdot 7 \cdot 227 \)

Automorphism group

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

Rational points

Known points
\((1 : 0 : 0)\)	\((1 : -1 : 0)\)	\((0 : 0 : 1)\)	\((0 : -1 : 1)\)	\((-1 : 1 : 2)\)	\((2 : -3 : 1)\)
\((-1 : -4 : 2)\)	\((-2 : 0 : 5)\)	\((2 : -8 : 1)\)	\((-1 : -63 : 6)\)	\((-2 : -67 : 5)\)	\((-1 : -116 : 6)\)

Number of rational Weierstrass points: \(0\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

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

Generator	$D_0$						Height	Order
\((-1 : 1 : 2) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\)	\(x (2x + z)\)	\(=\)	\(0,\)	\(4y\)	\(=\)	\(-9xz^2 - 4z^3\)	\(0.130426\)	\(\infty\)
\((-1 : 1 : 2) - (1 : 0 : 0)\)	\(z (2x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-x^3\)	\(0.060770\)	\(\infty\)

2-torsion field: 6.4.508480.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(2\)
Mordell-Weil rank:	\(2\)
2-Selmer rank:	\(2\)
Regulator:	\( 0.007897 \)
Real period:	\( 17.93444 \)
Tamagawa product:	\( 3 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 0.424929 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(5\)	\(1\)	\(3\)	\(3\)	\(( 1 - T )( 1 + 4 T + 5 T^{2} )\)
\(7\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 + 2 T + 7 T^{2} )\)
\(227\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 - 4 T + 227 T^{2} )\)

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 \(\Q\).