Genus 2 curve 91771.a.91771.1

• Label: 91771.a.91771.1
• Conductor: $91771$
• Discriminant: $-91771$
• 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

$y^2 + y = x^5 + x^4 - x^2 - x$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(91771\)	\(=\)	\( 91771 \)
Discriminant:	\( \Delta \)	\(=\)	\(-91771\)	\(=\)	\( -91771 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(96\)	\(=\)	\( 2^{5} \cdot 3 \)
\( I_4 \)	\(=\)	\(-2160\)	\(=\)	\( - 2^{4} \cdot 3^{3} \cdot 5 \)
\( I_6 \)	\(=\)	\(-15984\)	\(=\)	\( - 2^{4} \cdot 3^{3} \cdot 37 \)
\( I_{10} \)	\(=\)	\(367084\)	\(=\)	\( 2^{2} \cdot 91771 \)

Automorphism group

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

Rational points

Known points
\((1 : 0 : 0)\)	\((0 : 0 : 1)\)	\((-1 : 0 : 1)\)	\((0 : -1 : 1)\)	\((1 : 0 : 1)\)	\((-1 : -1 : 1)\)
\((1 : -1 : 1)\)	\((2 : 6 : 1)\)	\((2 : -7 : 1)\)	\((-3 : 14 : 4)\)	\((4 : 35 : 1)\)	\((4 : -36 : 1)\)
\((-3 : -78 : 4)\)	\((1 : -105 : 9)\)	\((1 : -624 : 9)\)

Number of rational Weierstrass points: \(1\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

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

Generator	$D_0$						Height	Order
\((-1 : -1 : 1) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)\)	\((x - z) (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-z^3\)	\(0.685044\)	\(\infty\)
\((-1 : -1 : 1) - (1 : 0 : 0)\)	\(x + z\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-z^3\)	\(0.532129\)	\(\infty\)
\((0 : -1 : 1) - (1 : 0 : 0)\)	\(x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-z^3\)	\(0.217433\)	\(\infty\)

2-torsion field: 5.3.1468336.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(3\)   (upper bound)
Mordell-Weil rank:	\(3\)
2-Selmer rank:	\(3\)
Regulator:	\( 0.065991 \)
Real period:	\( 13.22064 \)
Tamagawa product:	\( 1 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 0.872444 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(91771\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 + 235 T + 91771 T^{2} )\)

Galois representations

The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) except those listed.

Prime \(\ell\)	mod-\(\ell\) image	Is torsion prime?
\(2\)	2.6.1	no

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 \(\Q\).