Genus 2 curve 202500.a.405000.1

• Label: 202500.a.405000.1
• Conductor: $202500$
• Discriminant: $-405000$
• Mordell-Weil group: \(\Z \oplus \Z\)
• Sato-Tate group: $E_3$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\R)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\Q)\)
• \(\End(J) \otimes \Q\): \(\mathsf{CM}\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

Minimal equation

$y^2 + (x^3 + x + 1)y = -2x^5 + 6x^4 - 6x^3$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(202500\)	\(=\)	\( 2^{2} \cdot 3^{4} \cdot 5^{4} \)
Discriminant:	\( \Delta \)	\(=\)	\(-405000\)	\(=\)	\( - 2^{3} \cdot 3^{4} \cdot 5^{4} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(804\)	\(=\)	\( 2^{2} \cdot 3 \cdot 67 \)
\( I_4 \)	\(=\)	\(72225\)	\(=\)	\( 3^{3} \cdot 5^{2} \cdot 107 \)
\( I_6 \)	\(=\)	\(13647825\)	\(=\)	\( 3^{3} \cdot 5^{2} \cdot 20219 \)
\( I_{10} \)	\(=\)	\(-51840000\)	\(=\)	\( - 2^{10} \cdot 3^{4} \cdot 5^{4} \)

Automorphism group

\(\mathrm{Aut}(X)\)	\(\simeq\)	$C_6$
\(\mathrm{Aut}(X_{\overline{\Q}})\)	\(\simeq\)	$D_6$

Rational points

Known points
\((1 : 0 : 0)\)	\((1 : -1 : 0)\)	\((0 : 0 : 1)\)	\((0 : -1 : 1)\)	\((1 : -1 : 1)\)	\((1 : -2 : 1)\)
\((5 : -28 : 2)\)	\((-2 : 48 : 3)\)	\((-2 : -49 : 3)\)	\((3 : -65 : 5)\)	\((5 : -125 : 2)\)	\((3 : -162 : 5)\)

Number of rational Weierstrass points: \(0\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z \oplus \Z\)

Generator	$D_0$						Height	Order
\((0 : -1 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\)	\(x (x - z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-z^3\)	\(0.199552\)	\(\infty\)
\((1 : -1 : 1) - (1 : 0 : 0)\)	\(z (x - z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-x^3\)	\(0.199552\)	\(\infty\)

2-torsion field: 6.0.25920000.1

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(2\)
Mordell-Weil rank:	\(2\)
2-Selmer rank:	\(2\)
Regulator:	\( 0.029865 \)
Real period:	\( 18.48871 \)
Tamagawa product:	\( 3 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 1.656551 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

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

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?
\(2\)	2.40.3	no
\(3\)	3.960.8	no

Sato-Tate group

\(\mathrm{ST}\)	\(\simeq\)	$E_3$
\(\mathrm{ST}^0\)	\(\simeq\)	\(\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) \(\Q(\zeta_{9})^+\) with defining polynomial:
  \(x^{3} - 3 x - 1\)

Decomposes up to isogeny as the square of the elliptic curve isogeny class:
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = -\frac{35775}{16} b^{2} + \frac{7155}{8} b + \frac{107325}{16}\)
  \(g_6 = \frac{11533995}{64} b^{2} - \frac{1048545}{16} b - \frac{1048545}{2}\)
   Conductor norm: 125000

Endomorphisms of the Jacobian

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

Endomorphism ring over \(\Q\):

\(\End (J_{})\)	\(\simeq\)	\(\Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{-3}) \)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\C\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\zeta_{9})^+\) with defining polynomial \(x^{3} - 3 x - 1\)

Not of \(\GL_2\)-type over \(\overline{\Q}\)

Endomorphism ring over \(\overline{\Q}\):

\(\End (J_{\overline{\Q}})\)	\(\simeq\)	an Eichler order of index \(3\) in a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)	\(\simeq\)	\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{\overline{\Q}}) \otimes \R\)	\(\simeq\)	\(\mathrm{M}_2 (\R)\)