Genus 2 curve 1125.a.151875.1

• Label: 1125.a.151875.1
• Conductor: $1125$
• Discriminant: $-151875$
• Mordell-Weil group: \(\Z/{2}\Z \oplus \Z/{2}\Z\)
• Sato-Tate group: $\mathrm{SU}(2)\times\mathrm{SU}(2)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\R \times \R\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\Q \times \Q\)
• \(\End(J) \otimes \Q\): \(\Q \times \Q\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

Minimal equation

$y^2 + xy = 15x^5 + 50x^4 + 55x^3 + 22x^2 + 3x$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(1125\)	\(=\)	\( 3^{2} \cdot 5^{3} \)
Discriminant:	\( \Delta \)	\(=\)	\(-151875\)	\(=\)	\( - 3^{5} \cdot 5^{4} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(8600\)	\(=\)	\( 2^{3} \cdot 5^{2} \cdot 43 \)
\( I_4 \)	\(=\)	\(612100\)	\(=\)	\( 2^{2} \cdot 5^{2} \cdot 6121 \)
\( I_6 \)	\(=\)	\(1556297975\)	\(=\)	\( 5^{2} \cdot 62251919 \)
\( I_{10} \)	\(=\)	\(-607500\)	\(=\)	\( - 2^{2} \cdot 3^{5} \cdot 5^{4} \)

Automorphism group

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

Rational points

All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (-4 : 18 : 3)\)

Number of rational Weierstrass points: \(3\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

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

Generator	$D_0$						Height	Order
\((-4 : 18 : 3) - (1 : 0 : 0)\)	\(3x + 4z\)	\(=\)	\(0,\)	\(3y\)	\(=\)	\(2z^3\)	\(0\)	\(2\)
\((0 : 0 : 1) - (1 : 0 : 0)\)	\(x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0\)	\(2\)

2-torsion field: 3.1.300.1

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(2\)
Regulator:	\( 1 \)
Real period:	\( 1.964401 \)
Tamagawa product:	\( 4 \)
Torsion order:	\( 4 \)
Leading coefficient:	\( 0.491100 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(3\)	\(2\)	\(5\)	\(2\)	\(( 1 + T )^{2}\)
\(5\)	\(3\)	\(4\)	\(2\)	\(1 - T\)

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?
\(2\)	2.120.3	yes
\(3\)	3.80.4	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

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 75.c

Endomorphisms of the Jacobian

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

Endomorphism ring over \(\Q\):

\(\End (J_{})\)	\(\simeq\)	an order of index \(3\) 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 \(\Q\).