Genus 2 curve 14529.a.392283.1

• Label: 14529.a.392283.1
• Conductor: $14529$
• Discriminant: $-392283$
• Mordell-Weil group: \(\Z \oplus \Z\)
• 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

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

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(14529\)	\(=\)	\( 3 \cdot 29 \cdot 167 \)
Discriminant:	\( \Delta \)	\(=\)	\(-392283\)	\(=\)	\( - 3^{4} \cdot 29 \cdot 167 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(340\)	\(=\)	\( 2^{2} \cdot 5 \cdot 17 \)
\( I_4 \)	\(=\)	\(-8471\)	\(=\)	\( - 43 \cdot 197 \)
\( I_6 \)	\(=\)	\(-1484651\)	\(=\)	\( - 7^{2} \cdot 41 \cdot 739 \)
\( I_{10} \)	\(=\)	\(-50212224\)	\(=\)	\( - 2^{7} \cdot 3^{4} \cdot 29 \cdot 167 \)

Automorphism group

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

Rational points

Known points
\((1 : 0 : 0)\)	\((1 : -1 : 0)\)	\((0 : 0 : 1)\)	\((0 : -1 : 1)\)	\((1 : -1 : 1)\)	\((-2 : 2 : 1)\)
\((-1 : -2 : 2)\)	\((-2 : 5 : 1)\)	\((-1 : -5 : 2)\)	\((-3 : 5 : 1)\)	\((-3 : 21 : 1)\)

Number of rational Weierstrass points: \(1\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

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

Generator	$D_0$						Height	Order
\((-2 : 2 : 1) - (1 : -1 : 0)\)	\(z (x + 2z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(2z^3\)	\(0.327871\)	\(\infty\)
\((-2 : 2 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\)	\(x (x + 2z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-xz^2\)	\(0.033452\)	\(\infty\)

2-torsion field: 5.3.77488.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(2\)
Mordell-Weil rank:	\(2\)
2-Selmer rank:	\(2\)
Regulator:	\( 0.010794 \)
Real period:	\( 12.89199 \)
Tamagawa product:	\( 4 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 0.556641 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(3\)	\(1\)	\(4\)	\(4\)	\(( 1 - T )( 1 + 2 T + 3 T^{2} )\)
\(29\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 + 7 T + 29 T^{2} )\)
\(167\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 + 2 T + 167 T^{2} )\)

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 \(\Q\).