Genus 2 curve 249478.a.498956.1

• Label: 249478.a.498956.1
• Conductor: $249478$
• Discriminant: $-498956$
• 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

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

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(249478\)	\(=\)	\( 2 \cdot 124739 \)
Discriminant:	\( \Delta \)	\(=\)	\(-498956\)	\(=\)	\( - 2^{2} \cdot 124739 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(1832\)	\(=\)	\( 2^{3} \cdot 229 \)
\( I_4 \)	\(=\)	\(33760\)	\(=\)	\( 2^{5} \cdot 5 \cdot 211 \)
\( I_6 \)	\(=\)	\(22104779\)	\(=\)	\( 197 \cdot 112207 \)
\( I_{10} \)	\(=\)	\(-1995824\)	\(=\)	\( - 2^{4} \cdot 124739 \)

Automorphism group

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

Rational points

Known points
\((1 : 0 : 0)\)	\((0 : 0 : 1)\)	\((0 : -1 : 1)\)	\((1 : 1 : 1)\)	\((-2 : -2 : 1)\)	\((1 : -3 : 1)\)
\((-2 : -3 : 1)\)	\((1 : -4 : 2)\)	\((1 : -6 : 2)\)	\((2 : 9 : 1)\)	\((-3 : -12 : 2)\)	\((2 : -14 : 1)\)
\((-3 : -14 : 2)\)	\((9 : 522 : 2)\)	\((9 : -692 : 2)\)

Number of rational Weierstrass points: \(1\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

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

Generator	$D_0$						Height	Order
\((1 : -6 : 2) - (1 : 0 : 0)\)	\(2x - z\)	\(=\)	\(0,\)	\(4y\)	\(=\)	\(-3z^3\)	\(0.788464\)	\(\infty\)
\((0 : -1 : 1) - (1 : 0 : 0)\)	\(x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-z^3\)	\(0.362335\)	\(\infty\)
\(D_0 - 2 \cdot(1 : 0 : 0)\)	\(x^2 + xz - z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(xz^2 - z^3\)	\(0.268875\)	\(\infty\)

2-torsion field: 5.3.498956.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(3\)   (upper bound)
Mordell-Weil rank:	\(3\)
2-Selmer rank:	\(3\)
Regulator:	\( 0.066971 \)
Real period:	\( 12.89921 \)
Tamagawa product:	\( 2 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 1.727764 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(1\)	\(2\)	\(2\)	\(( 1 + T )( 1 + 2 T^{2} )\)
\(124739\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 + 225 T + 124739 T^{2} )\)

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 \(\Q\).