Genus 2 curve 7225.b.614125.1

• Label: 7225.b.614125.1
• Conductor: $7225$
• Discriminant: $-614125$
• Mordell-Weil group: \(\Z/{2}\Z\)
• 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

$y^2 + (x^3 + 1)y = -x^6 - 5x^5 + 67x^3 + 176x^2 + 155x + 32$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(7225\)	\(=\)	\( 5^{2} \cdot 17^{2} \)
Discriminant:	\( \Delta \)	\(=\)	\(-614125\)	\(=\)	\( - 5^{3} \cdot 17^{3} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(17140\)	\(=\)	\( 2^{2} \cdot 5 \cdot 857 \)
\( I_4 \)	\(=\)	\(19045441\)	\(=\)	\( 383 \cdot 49727 \)
\( I_6 \)	\(=\)	\(73673450257\)	\(=\)	\( 11 \cdot 199 \cdot 281 \cdot 119773 \)
\( I_{10} \)	\(=\)	\(-78608000\)	\(=\)	\( - 2^{7} \cdot 5^{3} \cdot 17^{3} \)

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 $\Q_{17}$.

Mordell-Weil group of the Jacobian

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

Generator	$D_0$						Height	Order
\(D_0 - D_\infty\)	\(3x^2 + 11xz + 3z^2\)	\(=\)	\(0,\)	\(9y\)	\(=\)	\(-56xz^2 - 21z^3\)	\(0\)	\(2\)

2-torsion field: 8.0.46240000.1

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(2\)
Regulator:	\( 1 \)
Real period:	\( 1.174397 \)
Tamagawa product:	\( 2 \)
Torsion order:	\( 2 \)
Leading coefficient:	\( 1.174397 \)
Analytic order of Ш:	\( 2 \)   (rounded)
Order of Ш:	twice a square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(5\)	\(2\)	\(3\)	\(2\)	\(( 1 - T )^{2}\)
\(17\)	\(2\)	\(3\)	\(1\)	\(( 1 + T )^{2}\)

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?
\(2\)	2.45.1	yes
\(3\)	3.640.4	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 \(\Q\):

\(\End (J_{})\)	\(\simeq\)	\(\Z [\sqrt{3}]\)
\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{3}) \)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\R \times \R\)

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