Genus 2 curve 2563.a.2563.1

• Label: 2563.a.2563.1
• Conductor: $2563$
• Discriminant: $2563$
• Mordell-Weil group: \(\Z\)
• 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

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

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(2563\)	\(=\)	\( 11 \cdot 233 \)
Discriminant:	\( \Delta \)	\(=\)	\(2563\)	\(=\)	\( 11 \cdot 233 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(212\)	\(=\)	\( 2^{2} \cdot 53 \)
\( I_4 \)	\(=\)	\(-6023\)	\(=\)	\( - 19 \cdot 317 \)
\( I_6 \)	\(=\)	\(-240627\)	\(=\)	\( - 3 \cdot 80209 \)
\( I_{10} \)	\(=\)	\(328064\)	\(=\)	\( 2^{7} \cdot 11 \cdot 233 \)

Automorphism group

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

Rational points

All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (-1 : 0 : 1),\, (0 : -1 : 1),\, (-1 : -1 : 1),\, (-3 : 12 : 1),\, (-3 : -19 : 1)\)

Number of rational Weierstrass points: \(1\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z\)

Generator	$D_0$						Height	Order
\((-1 : 0 : 1) + (0 : -1 : 1) - 2 \cdot(1 : 0 : 0)\)	\(x (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-xz^2 - z^3\)	\(0.019414\)	\(\infty\)

2-torsion field: 5.1.41008.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(1\)
Mordell-Weil rank:	\(1\)
2-Selmer rank:	\(1\)
Regulator:	\( 0.019414 \)
Real period:	\( 16.67202 \)
Tamagawa product:	\( 1 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 0.323686 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(11\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 + 5 T + 11 T^{2} )\)
\(233\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 - 3 T + 233 T^{2} )\)

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 \(\Q\).