Genus 2 curve 143313.a.429939.1

• Label: 143313.a.429939.1
• Conductor: $143313$
• Discriminant: $-429939$
• 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

$y^2 + x^3y = x^5 - 5x^3 - 5x^2 + 5x + 6$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(143313\)	\(=\)	\( 3 \cdot 23 \cdot 31 \cdot 67 \)
Discriminant:	\( \Delta \)	\(=\)	\(-429939\)	\(=\)	\( - 3^{2} \cdot 23 \cdot 31 \cdot 67 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(40\)	\(=\)	\( 2^{3} \cdot 5 \)
\( I_4 \)	\(=\)	\(-6080\)	\(=\)	\( - 2^{6} \cdot 5 \cdot 19 \)
\( I_6 \)	\(=\)	\(642680\)	\(=\)	\( 2^{3} \cdot 5 \cdot 16067 \)
\( I_{10} \)	\(=\)	\(1719756\)	\(=\)	\( 2^{2} \cdot 3^{2} \cdot 23 \cdot 31 \cdot 67 \)

Automorphism group

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

Rational points

Known points
\((1 : 0 : 0)\)	\((1 : -1 : 0)\)	\((-1 : 0 : 1)\)	\((-1 : 1 : 1)\)	\((1 : 1 : 1)\)	\((1 : -2 : 1)\)
\((2 : -2 : 1)\)	\((-2 : 4 : 1)\)	\((2 : -6 : 1)\)	\((-3 : 9 : 1)\)	\((-1 : -13 : 2)\)	\((-1 : 14 : 2)\)
\((-3 : 18 : 1)\)	\((-5 : 47 : 2)\)	\((-5 : 78 : 2)\)

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
\((-5 : 47 : 2) + (1 : -2 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\)	\((x - z) (2x + 5z)\)	\(=\)	\(0,\)	\(4y\)	\(=\)	\(-9xz^2 + z^3\)	\(1.315183\)	\(\infty\)
\((-1 : 0 : 1) - (1 : -1 : 0)\)	\(z (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0.535576\)	\(\infty\)
\((-2 : 4 : 1) - (1 : -1 : 0)\)	\(z (x + 2z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(4z^3\)	\(0.064949\)	\(\infty\)

2-torsion field: 5.3.764336.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(3\)   (upper bound)
Mordell-Weil rank:	\(3\)
2-Selmer rank:	\(3\)
Regulator:	\( 0.043807 \)
Real period:	\( 13.78651 \)
Tamagawa product:	\( 2 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 1.207900 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(3\)	\(1\)	\(2\)	\(2\)	\(( 1 + T )( 1 + T + 3 T^{2} )\)
\(23\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 + 9 T + 23 T^{2} )\)
\(31\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 + 31 T^{2} )\)
\(67\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 + 3 T + 67 T^{2} )\)

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 \(\Q\).