Genus 2 curve 990.a.8910.1

• Label: 990.a.8910.1
• Conductor: $990$
• Discriminant: $8910$
• Mordell-Weil group: \(\Z/{2}\Z \oplus \Z/{4}\Z\)
• Sato-Tate group: $\mathrm{SU}(2)\times\mathrm{SU}(2)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\R \times \R\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\Q \times \Q\)
• \(\End(J) \otimes \Q\): \(\Q \times \Q\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

Minimal equation

$y^2 + (x^2 + x)y = 3x^5 + 4x^4 + 7x^3 + 4x^2 + 3x$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(990\)	\(=\)	\( 2 \cdot 3^{2} \cdot 5 \cdot 11 \)
Discriminant:	\( \Delta \)	\(=\)	\(8910\)	\(=\)	\( 2 \cdot 3^{4} \cdot 5 \cdot 11 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(3268\)	\(=\)	\( 2^{2} \cdot 19 \cdot 43 \)
\( I_4 \)	\(=\)	\(252577\)	\(=\)	\( 13 \cdot 19429 \)
\( I_6 \)	\(=\)	\(318023313\)	\(=\)	\( 3 \cdot 43 \cdot 311 \cdot 7927 \)
\( I_{10} \)	\(=\)	\(1140480\)	\(=\)	\( 2^{8} \cdot 3^{4} \cdot 5 \cdot 11 \)

Automorphism group

\(\mathrm{Aut}(X)\)	\(\simeq\)	$C_2^2$
\(\mathrm{Aut}(X_{\overline{\Q}})\)	\(\simeq\)	$C_2^2$

Rational points

All points: \((1 : 0 : 0),\, (0 : 0 : 1)\)

Number of rational Weierstrass points: \(2\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z/{2}\Z \oplus \Z/{4}\Z\)

Generator	$D_0$						Height	Order
\(D_0 - 2 \cdot(1 : 0 : 0)\)	\(3x^2 + 2xz + 3z^2\)	\(=\)	\(0,\)	\(6y\)	\(=\)	\(-xz^2 + 3z^3\)	\(0\)	\(2\)
\(D_0 - 2 \cdot(1 : 0 : 0)\)	\(3x^2 + xz + 3z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0\)	\(4\)

2-torsion field: \(\Q(\sqrt{-2}, \sqrt{-55})\)

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(2\)
Regulator:	\( 1 \)
Real period:	\( 6.174936 \)
Tamagawa product:	\( 4 \)
Torsion order:	\( 8 \)
Leading coefficient:	\( 0.385933 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	Root number*	L-factor	Cluster picture	Tame reduction?
\(2\)	\(1\)	\(1\)	\(1\)	\(-1^*\)	\(( 1 - T )( 1 + T + 2 T^{2} )\)		yes
\(3\)	\(2\)	\(4\)	\(4\)	\(1\)	\(( 1 + T )^{2}\)		yes
\(5\)	\(1\)	\(1\)	\(1\)	\(-1\)	\(( 1 - T )( 1 - 2 T + 5 T^{2} )\)		yes
\(11\)	\(1\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 + 4 T + 11 T^{2} )\)		yes

Galois representations

For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.

For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not all of $\GSp(4,\F_\ell)$.

Prime \(\ell\)	mod-\(\ell\) image	Is torsion prime?
\(2\)	2.180.3	yes
\(3\)	3.90.1	no

Sato-Tate group

\(\mathrm{ST}\)	\(\simeq\)	$\mathrm{SU}(2)\times\mathrm{SU}(2)$
\(\mathrm{ST}^0\)	\(\simeq\)	\(\mathrm{SU}(2)\times\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  Elliptic curve isogeny class 15.a
  Elliptic curve isogeny class 66.b

Endomorphisms of the Jacobian

Of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)	\(\simeq\)	an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q\) \(\times\) \(\Q\)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\R \times \R\)

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).