Genus 2 curve 27225.a.81675.1

• Label: 27225.a.81675.1
• Conductor: $27225$
• Discriminant: $-81675$
• Mordell-Weil group: \(\Z \oplus \Z \oplus \Z/{2}\Z\)
• Sato-Tate group: $\mathrm{SU}(2)\times\mathrm{SU}(2)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\R \times \R\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathsf{RM}\)
• \(\End(J) \otimes \Q\): \(\mathsf{RM}\)
• \(\overline{\Q}\)-simple: yes
• \(\mathrm{GL}_2\)-type: yes

Minimal equation

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

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(27225\)	\(=\)	\( 3^{2} \cdot 5^{2} \cdot 11^{2} \)
Discriminant:	\( \Delta \)	\(=\)	\(-81675\)	\(=\)	\( - 3^{3} \cdot 5^{2} \cdot 11^{2} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(2060\)	\(=\)	\( 2^{2} \cdot 5 \cdot 103 \)
\( I_4 \)	\(=\)	\(56521\)	\(=\)	\( 29 \cdot 1949 \)
\( I_6 \)	\(=\)	\(34621947\)	\(=\)	\( 3^{2} \cdot 31^{2} \cdot 4003 \)
\( I_{10} \)	\(=\)	\(10454400\)	\(=\)	\( 2^{7} \cdot 3^{3} \cdot 5^{2} \cdot 11^{2} \)

Automorphism group

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

Rational points

Known points
\((1 : 0 : 0)\)	\((1 : -1 : 0)\)	\((0 : 1 : 1)\)	\((0 : -2 : 1)\)	\((-1 : -2 : 1)\)	\((-1 : 2 : 1)\)
\((7 : 1321 : 8)\)	\((7 : -2176 : 8)\)

Number of rational Weierstrass points: \(0\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

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

Generator	$D_0$						Height	Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\)	\(2x^2 + 2xz + 3z^2\)	\(=\)	\(0,\)	\(2y\)	\(=\)	\(xz^2 - z^3\)	\(0.393627\)	\(\infty\)
\((0 : -2 : 1) - (1 : -1 : 0)\)	\(z x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-2z^3\)	\(0.365989\)	\(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\)	\(x^2 + xz + z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-z^3\)	\(0\)	\(2\)

2-torsion field: 8.0.741200625.1

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(2\)
Mordell-Weil rank:	\(2\)
2-Selmer rank:	\(3\)
Regulator:	\( 0.143872 \)
Real period:	\( 9.543118 \)
Tamagawa product:	\( 2 \)
Torsion order:	\( 2 \)
Leading coefficient:	\( 0.686496 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(3\)	\(2\)	\(3\)	\(2\)	\(( 1 + T )^{2}\)
\(5\)	\(2\)	\(2\)	\(1\)	\(( 1 + T )^{2}\)
\(11\)	\(2\)	\(2\)	\(1\)	\(( 1 + T )^{2}\)

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?
\(2\)	2.90.5	yes
\(3\)	3.72.2	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

Simple over \(\overline{\Q}\)

Endomorphisms of the Jacobian

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

Endomorphism ring over \(\Q\):

\(\End (J_{})\)	\(\simeq\)	\(\Z [\sqrt{2}]\)
\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{2}) \)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\R \times \R\)

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