Genus 2 Curve 3319.a.3319.1

• Label: 3319.a.3319.1
• Conductor: 3319
• Discriminant: 3319
• Mordell-Weil group: \(\Z \times \Z\)
• Sato-Tate group: $\mathrm{USp}(4)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\R\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\Q\)
• \(\overline{\Q}\)-simple: yes
• \(\mathrm{GL}_2\)-type: no

Minimal equation

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

(, )

Invariants

Conductor:	\( N \)	=	\(3319\)	=	\( 3319 \)
Discriminant:	\( \Delta \)	=	\(3319\)	=	\( 3319 \)

Igusa-Clebsch invariants

\( I_2 \)	=	\(136\)	=	\( 2^{3} \cdot 17 \)
\( I_4 \)	=	\(14692\)	=	\( 2^{2} \cdot 3673 \)
\( I_6 \)	=	\(304744\)	=	\( 2^{3} \cdot 11 \cdot 3463 \)
\( I_{10} \)	=	\(13594624\)	=	\( 2^{12} \cdot 3319 \)

Automorphism group

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

Rational points

Known points
\((1 : 0 : 0)\)	\((1 : -1 : 0)\)	\((0 : 0 : 1)\)	\((-1 : 0 : 1)\)	\((0 : -1 : 1)\)	\((-1 : 1 : 1)\)
\((1 : 1 : 1)\)	\((-2 : 1 : 1)\)	\((-1 : -1 : 2)\)	\((-1 : -2 : 2)\)	\((1 : -4 : 1)\)	\((-2 : 8 : 1)\)

Number of rational Weierstrass points: \(0\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian:

Group structure: \(\Z \times \Z\)

Generator	$D_0$						Height	Order
\((-1 : 0 : 1) - (1 : -1 : 0)\)	\(z (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0.205416\)	\(\infty\)
\((-1 : 0 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\)	\(x (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0.036927\)	\(\infty\)

2-torsion field: 6.2.212416.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(2\)
Mordell-Weil rank:	\(2\)
2-Selmer rank:	\(2\)
Regulator:	\( 0.007585 \)
Real period:	\( 25.35935 \)
Tamagawa product:	\( 1 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 0.192363 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor
\(3319\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 - 74 T + 3319 T^{2} )\)

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 \(\Q\).