Genus 2 curve 1225.a.6125.1

• Label: 1225.a.6125.1
• Conductor: $1225$
• Discriminant: $6125$
• Mordell-Weil group: \(\Z/{8}\Z\)
• 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

The Jacobian of this curve is isogenous to that of the quotient of the modular curve $X_0(35)$ by the Atkin-Lehner involution $w_7$ , which has discriminant $5^77^3$.

Minimal equation

$y^2 + (x^3 + x^2)y = 2x^3 + x^2 + x + 2$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(1225\)	\(=\)	\( 5^{2} \cdot 7^{2} \)
Discriminant:	\( \Delta \)	\(=\)	\(6125\)	\(=\)	\( 5^{3} \cdot 7^{2} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(320\)	\(=\)	\( 2^{6} \cdot 5 \)
\( I_4 \)	\(=\)	\(14344\)	\(=\)	\( 2^{3} \cdot 11 \cdot 163 \)
\( I_6 \)	\(=\)	\(962481\)	\(=\)	\( 3 \cdot 13 \cdot 23 \cdot 29 \cdot 37 \)
\( I_{10} \)	\(=\)	\(-24500\)	\(=\)	\( - 2^{2} \cdot 5^{3} \cdot 7^{2} \)

Automorphism group

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

Rational points

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

Number of rational Weierstrass points: \(1\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z/{8}\Z\)

Generator	$D_0$						Height	Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\)	\(2x^2 - xz + 2z^2\)	\(=\)	\(0,\)	\(4y\)	\(=\)	\(xz^2 + 6z^3\)	\(0\)	\(8\)

2-torsion field: 6.2.98000.1

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(1\)
Regulator:	\( 1 \)
Real period:	\( 11.92789 \)
Tamagawa product:	\( 2 \)
Torsion order:	\( 8 \)
Leading coefficient:	\( 0.372746 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(5\)	\(2\)	\(3\)	\(2\)	\(( 1 - T )^{2}\)
\(7\)	\(2\)	\(2\)	\(1\)	\(( 1 + T )^{2}\)

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?
\(2\)	2.60.1	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 \(\Q\):

\(\End (J_{})\)	\(\simeq\)	\(\Z [\frac{1 + \sqrt{17}}{2}]\)
\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{17}) \)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\R \times \R\)

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