Genus 2 curve 1051.b.1051.1

• Label: 1051.b.1051.1
• Conductor: $1051$
• Discriminant: $-1051$
• Mordell-Weil group: \(\Z/{8}\Z\)
• 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

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

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(1051\)	\(=\)	\( 1051 \)
Discriminant:	\( \Delta \)	\(=\)	\(-1051\)	\(=\)	\( -1051 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(64\)	\(=\)	\( 2^{6} \)
\( I_4 \)	\(=\)	\(-200\)	\(=\)	\( - 2^{3} \cdot 5^{2} \)
\( I_6 \)	\(=\)	\(185\)	\(=\)	\( 5 \cdot 37 \)
\( I_{10} \)	\(=\)	\(4204\)	\(=\)	\( 2^{2} \cdot 1051 \)

Automorphism group

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

Rational points

All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (-1 : 0 : 1),\, (0 : -1 : 1)\)

Number of rational Weierstrass points: \(2\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z/{8}\Z\)

Generator	$D_0$						Height	Order
\((0 : 0 : 1) - (1 : 0 : 0)\)	\(x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0\)	\(8\)

2-torsion field: 4.2.4204.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(1\)
Regulator:	\( 1 \)
Real period:	\( 23.33171 \)
Tamagawa product:	\( 1 \)
Torsion order:	\( 8 \)
Leading coefficient:	\( 0.364558 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(1051\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 - 28 T + 1051 T^{2} )\)

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?
\(2\)	2.30.3	yes

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 \(\Q\).