Genus 2 curve 1519.a.1519.1

• Label: 1519.a.1519.1
• Conductor: $1519$
• Discriminant: $-1519$
• 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

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

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(1519\)	\(=\)	\( 7^{2} \cdot 31 \)
Discriminant:	\( \Delta \)	\(=\)	\(-1519\)	\(=\)	\( - 7^{2} \cdot 31 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(804\)	\(=\)	\( 2^{2} \cdot 3 \cdot 67 \)
\( I_4 \)	\(=\)	\(1953\)	\(=\)	\( 3^{2} \cdot 7 \cdot 31 \)
\( I_6 \)	\(=\)	\(517041\)	\(=\)	\( 3^{2} \cdot 7 \cdot 29 \cdot 283 \)
\( I_{10} \)	\(=\)	\(-194432\)	\(=\)	\( - 2^{7} \cdot 7^{2} \cdot 31 \)

Automorphism group

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

Rational points

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

Number of rational Weierstrass points: \(0\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z\)

Generator	$D_0$						Height	Order
\((-1 : 0 : 1) + (1 : -2 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\)	\((x - z) (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-xz^2 - z^3\)	\(0.011132\)	\(\infty\)

2-torsion field: 6.0.71528191.2

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(1\)
Mordell-Weil rank:	\(1\)
2-Selmer rank:	\(1\)
Regulator:	\( 0.011132 \)
Real period:	\( 23.93724 \)
Tamagawa product:	\( 1 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 0.266485 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(7\)	\(2\)	\(2\)	\(1\)	\(1 + T + 7 T^{2}\)
\(31\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 + 11 T + 31 T^{2} )\)

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?
\(2\)	2.40.2	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 \(\Q\).