Genus 2 curve 61553.a.61553.1

• Label: 61553.a.61553.1
• Conductor: $61553$
• Discriminant: $-61553$
• 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

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

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(61553\)	\(=\)	\( 61553 \)
Discriminant:	\( \Delta \)	\(=\)	\(-61553\)	\(=\)	\( -61553 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(260\)	\(=\)	\( 2^{2} \cdot 5 \cdot 13 \)
\( I_4 \)	\(=\)	\(20089\)	\(=\)	\( 20089 \)
\( I_6 \)	\(=\)	\(257469\)	\(=\)	\( 3 \cdot 19 \cdot 4517 \)
\( I_{10} \)	\(=\)	\(-7878784\)	\(=\)	\( - 2^{7} \cdot 61553 \)

Automorphism group

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

Rational points

Known points
\((1 : 0 : 0)\)	\((1 : -1 : 0)\)	\((0 : 0 : 1)\)	\((0 : -1 : 1)\)	\((-1 : -1 : 1)\)	\((1 : -1 : 1)\)
\((-2 : -1 : 1)\)	\((-1 : 1 : 2)\)	\((-1 : 2 : 1)\)	\((1 : -2 : 1)\)	\((-1 : -4 : 2)\)	\((3 : -6 : 1)\)
\((-2 : 10 : 1)\)	\((-2 : 18 : 3)\)	\((-2 : -19 : 3)\)	\((3 : -25 : 1)\)	\((24 : 266217 : 119)\)	\((24 : -2305064 : 119)\)

Number of rational Weierstrass points: \(0\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

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

Generator	$D_0$						Height	Order
\((-1 : -1 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\)	\(x (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(xz^2\)	\(0.597353\)	\(\infty\)
\((0 : 0 : 1) - (1 : -1 : 0)\)	\(z x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0.362615\)	\(\infty\)
\((0 : -1 : 1) - (1 : -1 : 0)\)	\(z x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-z^3\)	\(0.148079\)	\(\infty\)

2-torsion field: 6.4.3939392.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(3\)   (upper bound)
Mordell-Weil rank:	\(3\)
2-Selmer rank:	\(3\)
Regulator:	\( 0.029513 \)
Real period:	\( 20.66583 \)
Tamagawa product:	\( 1 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 0.609926 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(61553\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 - 246 T + 61553 T^{2} )\)

Galois representations

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

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 \(\Q\).