Genus 2 curve 688.a.2752.1

• Label: 688.a.2752.1
• Conductor: $688$
• Discriminant: $-2752$
• Mordell-Weil group: \(\Z/{20}\Z\)
• 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

$y^2 + y = 2x^5 - 5x^4 + 4x^3 - x$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(688\)	\(=\)	\( 2^{4} \cdot 43 \)
Discriminant:	\( \Delta \)	\(=\)	\(-2752\)	\(=\)	\( - 2^{6} \cdot 43 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(32\)	\(=\)	\( 2^{5} \)
\( I_4 \)	\(=\)	\(112\)	\(=\)	\( 2^{4} \cdot 7 \)
\( I_6 \)	\(=\)	\(-680\)	\(=\)	\( - 2^{3} \cdot 5 \cdot 17 \)
\( I_{10} \)	\(=\)	\(-344\)	\(=\)	\( - 2^{3} \cdot 43 \)

Automorphism group

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

Rational points

All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (1 : 0 : 1),\, (1 : -1 : 1),\, (1 : -4 : 2)\)

Number of rational Weierstrass points: \(2\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z/{20}\Z\)

Generator	$D_0$						Height	Order
\((0 : 0 : 1) - (1 : 0 : 0)\)	\(x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0\)	\(20\)

2-torsion field: 4.2.688.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(1\)
Regulator:	\( 1 \)
Real period:	\( 25.70729 \)
Tamagawa product:	\( 5 \)
Torsion order:	\( 20 \)
Leading coefficient:	\( 0.321341 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(4\)	\(6\)	\(5\)	\(1\)
\(43\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 - 4 T + 43 T^{2} )\)

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
\(5\)	not computed	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 \(\Q\).