Genus 2 curve 5655.a.84825.1

• Label: 5655.a.84825.1
• Conductor: $5655$
• Discriminant: $84825$
• Mordell-Weil group: \(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
• 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

$y^2 + (x^2 + x + 1)y = -5x^5 - 17x^4 + x^3 + 6x^2 - 3x$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(5655\)	\(=\)	\( 3 \cdot 5 \cdot 13 \cdot 29 \)
Discriminant:	\( \Delta \)	\(=\)	\(84825\)	\(=\)	\( 3^{2} \cdot 5^{2} \cdot 13 \cdot 29 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(18580\)	\(=\)	\( 2^{2} \cdot 5 \cdot 929 \)
\( I_4 \)	\(=\)	\(84985\)	\(=\)	\( 5 \cdot 23 \cdot 739 \)
\( I_6 \)	\(=\)	\(530502093\)	\(=\)	\( 3^{2} \cdot 11 \cdot 5358607 \)
\( I_{10} \)	\(=\)	\(10857600\)	\(=\)	\( 2^{7} \cdot 3^{2} \cdot 5^{2} \cdot 13 \cdot 29 \)

Automorphism group

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

Rational points

All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (1 : -42 : 4),\, (1 : -70 : 5),\, (1 : -85 : 5)\)

Number of rational Weierstrass points: \(2\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)

Generator	$D_0$						Height	Order
\((0 : -1 : 1) - (1 : 0 : 0)\)	\(x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-z^3\)	\(0.134006\)	\(\infty\)
\(D_0 - 2 \cdot(1 : 0 : 0)\)	\(5x^2 + 3xz - z^2\)	\(=\)	\(0,\)	\(5y\)	\(=\)	\(-xz^2 - 3z^3\)	\(0\)	\(2\)
\((1 : -42 : 4) - (1 : 0 : 0)\)	\(4x - z\)	\(=\)	\(0,\)	\(32y\)	\(=\)	\(-21z^3\)	\(0\)	\(2\)

2-torsion field: \(\Q(\sqrt{13}, \sqrt{29})\)

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(1\)
Mordell-Weil rank:	\(1\)
2-Selmer rank:	\(3\)
Regulator:	\( 0.134006 \)
Real period:	\( 13.33915 \)
Tamagawa product:	\( 4 \)
Torsion order:	\( 4 \)
Leading coefficient:	\( 0.446883 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(3\)	\(1\)	\(2\)	\(2\)	\(( 1 - T )( 1 + 2 T + 3 T^{2} )\)
\(5\)	\(1\)	\(2\)	\(2\)	\(( 1 + T )( 1 + 5 T^{2} )\)
\(13\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 - 6 T + 13 T^{2} )\)
\(29\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 + 6 T + 29 T^{2} )\)

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?
\(2\)	2.180.3	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 \(\Q\).