Genus 2 curve 143849.a.143849.1

• Label: 143849.a.143849.1
• Conductor: $143849$
• Discriminant: $143849$
• 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

$y^2 + (x^2 + x + 1)y = -x^5 + 3x^4 - 9x^2 + x + 6$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(143849\)	\(=\)	\( 19 \cdot 67 \cdot 113 \)
Discriminant:	\( \Delta \)	\(=\)	\(143849\)	\(=\)	\( 19 \cdot 67 \cdot 113 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(2964\)	\(=\)	\( 2^{2} \cdot 3 \cdot 13 \cdot 19 \)
\( I_4 \)	\(=\)	\(-11895\)	\(=\)	\( - 3 \cdot 5 \cdot 13 \cdot 61 \)
\( I_6 \)	\(=\)	\(-21353019\)	\(=\)	\( - 3 \cdot 29 \cdot 245437 \)
\( I_{10} \)	\(=\)	\(18412672\)	\(=\)	\( 2^{7} \cdot 19 \cdot 67 \cdot 113 \)

Automorphism group

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

Rational points

Known points
\((1 : 0 : 0)\)	\((-1 : 0 : 1)\)	\((1 : 0 : 1)\)	\((-1 : -1 : 1)\)	\((0 : 2 : 1)\)	\((0 : -3 : 1)\)
\((1 : -3 : 1)\)	\((2 : -3 : 1)\)	\((2 : -4 : 1)\)	\((-3 : 17 : 1)\)	\((-3 : -24 : 1)\)	\((-12 : 494 : 1)\)
\((-12 : -627 : 1)\)	\((12 : 214781 : 49)\)	\((12 : -368298 : 49)\)

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
\(D_0 - 2 \cdot(1 : 0 : 0)\)	\(x^2 - xz - z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-2xz^2\)	\(0.754034\)	\(\infty\)
\((2 : -3 : 1) - (1 : 0 : 0)\)	\(x - 2z\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-3z^3\)	\(0.561566\)	\(\infty\)
\((-1 : 0 : 1) + (2 : -3 : 1) - 2 \cdot(1 : 0 : 0)\)	\((x - 2z) (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-xz^2 - z^3\)	\(0.210027\)	\(\infty\)

2-torsion field: 5.1.2301584.2

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(3\)   (upper bound)
Mordell-Weil rank:	\(3\)
2-Selmer rank:	\(3\)
Regulator:	\( 0.078082 \)
Real period:	\( 15.25439 \)
Tamagawa product:	\( 1 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 1.191100 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(19\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 + 2 T + 19 T^{2} )\)
\(67\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 + 2 T + 67 T^{2} )\)
\(113\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 + 2 T + 113 T^{2} )\)

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 \(\Q\).