Genus 2 curve 1350.a.5400.1

• Label: 1350.a.5400.1
• Conductor: $1350$
• Discriminant: $5400$
• Mordell-Weil group: \(\Z/{12}\Z\)
• 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

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

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(1350\)	\(=\)	\( 2 \cdot 3^{3} \cdot 5^{2} \)
Discriminant:	\( \Delta \)	\(=\)	\(5400\)	\(=\)	\( 2^{3} \cdot 3^{3} \cdot 5^{2} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(1380\)	\(=\)	\( 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
\( I_4 \)	\(=\)	\(3969\)	\(=\)	\( 3^{4} \cdot 7^{2} \)
\( I_6 \)	\(=\)	\(1536129\)	\(=\)	\( 3^{2} \cdot 7 \cdot 37 \cdot 659 \)
\( I_{10} \)	\(=\)	\(691200\)	\(=\)	\( 2^{10} \cdot 3^{3} \cdot 5^{2} \)

Automorphism group

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

Rational points

All points: \((1 : 0 : 0),\, (-1 : -1 : 1),\, (-1 : 1 : 1),\, (-2 : -1 : 1)\)

Number of rational Weierstrass points: \(2\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z/{12}\Z\)

Generator	$D_0$						Height	Order
\((-1 : 1 : 1) - (1 : 0 : 0)\)	\(x + z\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(z^3\)	\(0\)	\(12\)

2-torsion field: 4.0.5400.1

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(1\)
Regulator:	\( 1 \)
Real period:	\( 22.24770 \)
Tamagawa product:	\( 3 \)
Torsion order:	\( 12 \)
Leading coefficient:	\( 0.463493 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(1\)	\(3\)	\(3\)	\(( 1 - T )( 1 + T + 2 T^{2} )\)
\(3\)	\(3\)	\(3\)	\(1\)	\(1 + T\)
\(5\)	\(2\)	\(2\)	\(1\)	\(( 1 - T )( 1 + T )\)

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?
\(2\)	2.90.3	yes
\(3\)	3.720.4	yes

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 90.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 \(\Q\).