Genus 2 curve 25600.f.512000.1

• Label: 25600.f.512000.1
• Conductor: $25600$
• Discriminant: $512000$
• Mordell-Weil group: \(\Z/{2}\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
• Sato-Tate group: $J(E_1)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\R)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\Q)\)
• \(\End(J) \otimes \Q\): \(\mathsf{RM}\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

Minimal equation

$y^2 = 2x^5 - 5x^4 - x^3 + 5x^2 + 2x$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(25600\)	\(=\)	\( 2^{10} \cdot 5^{2} \)
Discriminant:	\( \Delta \)	\(=\)	\(512000\)	\(=\)	\( 2^{12} \cdot 5^{3} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(566\)	\(=\)	\( 2 \cdot 283 \)
\( I_4 \)	\(=\)	\(2164\)	\(=\)	\( 2^{2} \cdot 541 \)
\( I_6 \)	\(=\)	\(432824\)	\(=\)	\( 2^{3} \cdot 7 \cdot 59 \cdot 131 \)
\( I_{10} \)	\(=\)	\(2000\)	\(=\)	\( 2^{4} \cdot 5^{3} \)

Automorphism group

\(\mathrm{Aut}(X)\)	\(\simeq\)	$C_2$
\(\mathrm{Aut}(X_{\overline{\Q}})\)	\(\simeq\)	$C_2^2$

Rational points

All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (-1 : 0 : 2),\, (2 : 0 : 1)\)

Number of rational Weierstrass points: \(4\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

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

Generator	$D_0$						Height	Order
\((0 : 0 : 1) - (1 : 0 : 0)\)	\(x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0\)	\(2\)
\((-1 : 0 : 2) + (0 : 0 : 1) - 2 \cdot(1 : 0 : 0)\)	\(x (2x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0\)	\(2\)
\((-1 : 0 : 2) + (2 : 0 : 1) - 2 \cdot(1 : 0 : 0)\)	\((x - 2z) (2x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0\)	\(2\)

2-torsion field: \(\Q(\sqrt{5}) \)

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(3\)
Regulator:	\( 1 \)
Real period:	\( 11.50149 \)
Tamagawa product:	\( 8 \)
Torsion order:	\( 8 \)
Leading coefficient:	\( 1.437687 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(10\)	\(12\)	\(4\)	\(1\)
\(5\)	\(2\)	\(3\)	\(2\)	\(( 1 - T )^{2}\)

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?
\(2\)	2.360.2	yes
\(3\)	3.1080.4	no

Sato-Tate group

\(\mathrm{ST}\)	\(\simeq\)	$J(E_1)$
\(\mathrm{ST}^0\)	\(\simeq\)	\(\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) \(\Q(\sqrt{-1}) \) with defining polynomial:
  \(x^{2} + 1\)

Decomposes up to isogeny as the square of the elliptic curve isogeny class:
  Elliptic curve isogeny class 2.0.4.1-1600.2-b

Endomorphisms of the Jacobian

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

Endomorphism ring over \(\Q\):

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

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{-1}) \) with defining polynomial \(x^{2} + 1\)

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

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

\(\End (J_{\overline{\Q}})\)	\(\simeq\)	a non-Eichler order of index \(8\) in a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)	\(\simeq\)	\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{\overline{\Q}}) \otimes \R\)	\(\simeq\)	\(\mathrm{M}_2 (\R)\)