Genus 2 curve 4482.a.4482.1

• Label: 4482.a.4482.1
• Conductor: $4482$
• Discriminant: $-4482$
• Mordell-Weil group: \(\Z/{5}\Z\)
• 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

$y^2 + xy = -3x^6 - 3x^5 - 9x^4 - 5x^3 - 8x^2 - 2x - 2$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(4482\)	\(=\)	\( 2 \cdot 3^{3} \cdot 83 \)
Discriminant:	\( \Delta \)	\(=\)	\(-4482\)	\(=\)	\( - 2 \cdot 3^{3} \cdot 83 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(8664\)	\(=\)	\( 2^{3} \cdot 3 \cdot 19^{2} \)
\( I_4 \)	\(=\)	\(7884\)	\(=\)	\( 2^{2} \cdot 3^{3} \cdot 73 \)
\( I_6 \)	\(=\)	\(22484025\)	\(=\)	\( 3^{2} \cdot 5^{2} \cdot 99929 \)
\( I_{10} \)	\(=\)	\(17928\)	\(=\)	\( 2^{3} \cdot 3^{3} \cdot 83 \)

Automorphism group

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

Rational points

This curve has no rational points.

Number of rational Weierstrass points: \(0\)

This curve is locally solvable except over $\R$.

Mordell-Weil group of the Jacobian

Group structure: \(\Z/{5}\Z\)

Generator	$D_0$						Height	Order
\(D_0 - D_\infty\)	\(x^2 + z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0\)	\(5\)

2-torsion field: 6.0.286848.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(1\)
Regulator:	\( 1 \)
Real period:	\( 7.242157 \)
Tamagawa product:	\( 1 \)
Torsion order:	\( 5 \)
Leading coefficient:	\( 0.579372 \)
Analytic order of Ш:	\( 2 \)   (rounded)
Order of Ш:	twice a square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 + 2 T + 2 T^{2} )\)
\(3\)	\(3\)	\(3\)	\(1\)	\(1 + T + 3 T^{2}\)
\(83\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 - 9 T + 83 T^{2} )\)

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?
\(3\)	3.80.2	no
\(5\)	not computed	yes

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 \(\Q\).