Genus 2 curve 25600.e.128000.1

• Label: 25600.e.128000.1
• Conductor: $25600$
• Discriminant: $-128000$
• Mordell-Weil group: \(\Z/{2}\Z \oplus \Z/{2}\Z\)
• Sato-Tate group: $E_4$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\R)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\Q)\)
• \(\End(J) \otimes \Q\): \(\mathsf{CM}\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

Minimal equation

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

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(25600\)	\(=\)	\( 2^{10} \cdot 5^{2} \)
Discriminant:	\( \Delta \)	\(=\)	\(-128000\)	\(=\)	\( - 2^{10} \cdot 5^{3} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(56\)	\(=\)	\( 2^{3} \cdot 7 \)
\( I_4 \)	\(=\)	\(-80\)	\(=\)	\( - 2^{4} \cdot 5 \)
\( I_6 \)	\(=\)	\(-260\)	\(=\)	\( - 2^{2} \cdot 5 \cdot 13 \)
\( I_{10} \)	\(=\)	\(500\)	\(=\)	\( 2^{2} \cdot 5^{3} \)

Automorphism group

\(\mathrm{Aut}(X)\)	\(\simeq\)	$C_4$
\(\mathrm{Aut}(X_{\overline{\Q}})\)	\(\simeq\)	$D_4$

Rational points

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

Number of rational Weierstrass points: \(2\)

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
\(D_0 - 2 \cdot(1 : 0 : 0)\)	\(x^2 - xz - z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0\)	\(2\)
\((0 : 0 : 1) - (1 : 0 : 0)\)	\(x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0\)	\(2\)

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

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(2\)
Regulator:	\( 1 \)
Real period:	\( 9.947770 \)
Tamagawa product:	\( 2 \)
Torsion order:	\( 4 \)
Leading coefficient:	\( 1.243471 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	Root number*	L-factor	Cluster picture	Tame reduction?
\(2\)	\(10\)	\(10\)	\(1\)	\(-1^*\)	\(1\)		no
\(5\)	\(2\)	\(3\)	\(2\)	\(-1\)	\(1 - 2 T + 5 T^{2}\)		yes

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?
\(2\)	2.180.3	yes
\(3\)	3.540.6	no

Sato-Tate group

\(\mathrm{ST}\)	\(\simeq\)	$E_4$
\(\mathrm{ST}^0\)	\(\simeq\)	\(\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) \(\Q(\zeta_{20})^+\) with defining polynomial:
  \(x^{4} - 5 x^{2} + 5\)

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
  \(g_4 = -\frac{32303200}{130321} b^{3} + \frac{61624800}{130321} b^{2} + \frac{44011200}{130321} b - \frac{84012560}{130321}\)
  \(g_6 = -\frac{530918514880}{47045881} b^{3} + \frac{1009835128000}{47045881} b^{2} + \frac{734310067200}{47045881} b - \frac{1396419152000}{47045881}\)
   Conductor norm: 4096

Endomorphisms of the Jacobian

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

Endomorphism ring over \(\Q\):

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

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\zeta_{20})^+\) with defining polynomial \(x^{4} - 5 x^{2} + 5\)

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

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

\(\End (J_{\overline{\Q}})\)	\(\simeq\)	a non-Eichler order of index \(4\) in a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)	\(\simeq\)	\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{\overline{\Q}}) \otimes \R\)	\(\simeq\)	\(\mathrm{M}_2 (\R)\)

Remainder of the endomorphism lattice by field

Over subfield \(F \simeq \) \(\Q(\sqrt{5}) \) with generator \(a^{2} - 2\) with minimal polynomial \(x^{2} - x - 1\):

\(\End (J_{F})\)	\(\simeq\)	\(\Z [\sqrt{-1}]\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{-1}) \)
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\C\)

  Sato Tate group: $E_2$
  Of \(\GL_2\)-type, simple