Genus 2 curve 102400.b.102400.1

• Label: 102400.b.102400.1
• Conductor: $102400$
• Discriminant: $-102400$
• Mordell-Weil group: $\Z \oplus \Z/{2}\Z$
• Sato-Tate group: $J(E_2)$
• End⁡(JQ‾)⊗R\End(J_{\overline{\Q}}) \otimes \R: $\mathrm{M}_2(\R)$
• End⁡(JQ‾)⊗Q\End(J_{\overline{\Q}}) \otimes \Q: $\mathrm{M}_2(\Q)$
• End⁡(J)⊗Q\End(J) \otimes \Q: $\Q$
• Q‾\overline{\Q}-simple: no
• GL2\mathrm{GL}_2-type: no

Minimal equation

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

(, )

Invariants

Conductor:	$N$	$=$	$102400$	$=$	$2^{12} \cdot 5^{2}$
Discriminant:	$\Delta$	$=$	$-102400$	$=$	$- 2^{12} \cdot 5^{2}$

Igusa-Clebsch invariants

$I_2$	$=$	$34$	$=$	$2 \cdot 17$
$I_4$	$=$	$-116$	$=$	$- 2^{2} \cdot 29$
$I_6$	$=$	$-424$	$=$	$- 2^{3} \cdot 53$
$I_{10}$	$=$	$400$	$=$	$2^{4} \cdot 5^{2}$

Automorphism group

$\mathrm{Aut}(X)$	$\simeq$	$C_2$
$\mathrm{Aut}(X_{\overline{\Q}})$	$\simeq$	$D_4$

Rational points

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

Number of rational Weierstrass points: $2$

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: $\Z \oplus \Z/{2}\Z$

Generator	$D_0$						Height	Order
$(-1 : -1 : 1) - (1 : 0 : 0)$	$x + z$	$=$	$0,$	$y$	$=$	$-z^3$	$0.797280$	$\infty$
$(0 : 0 : 1) - (1 : 0 : 0)$	$x$	$=$	$0,$	$y$	$=$	$0$	$0$	$2$

2-torsion field: 4.0.320.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	$1$
Mordell-Weil rank:	$1$
2-Selmer rank:	$2$
Regulator:	$0.797280$
Real period:	$12.08406$
Tamagawa product:	$1$
Torsion order:	$2$
Leading coefficient:	$2.408595$
Analytic order of Ш:	$1$   (rounded)
Order of Ш:	square

Local invariants

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

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?
$2$	2.90.3	yes
$3$	3.540.2	no

Sato-Tate group

$\mathrm{ST}$	$\simeq$	$J(E_2)$
$\mathrm{ST}^0$	$\simeq$	$\mathrm{SU}(2)$

Decomposition of the Jacobian

Splits over the number field $\Q (b) \simeq$ $\Q(\zeta_{8})$ with defining polynomial:
  $x^{4} + 1$

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
  $g_4 = 160 b^{2} - 48$
  $g_6 = 1792 b^{3} - 1152 b$
   Conductor norm: 160000

Endomorphisms of the Jacobian

Not of $\GL_2$-type over $\Q$

Endomorphism ring over $\Q$:

$\End (J_{})$	$\simeq$	$\Z$
$\End (J_{}) \otimes \Q$	$\simeq$	$\Q$
$\End (J_{}) \otimes \R$	$\simeq$	$\R$

Smallest field over which all endomorphisms are defined:
Galois number field $K = \Q (a) \simeq$ $\Q(\zeta_{8})$ with defining polynomial $x^{4} + 1$

Not of $\GL_2$-type over $\overline{\Q}$

Endomorphism ring over $\overline{\Q}$:

$\End (J_{\overline{\Q}})$	$\simeq$	a non-Eichler order of index $4$ 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 $F \simeq$ $\Q(\sqrt{-2})$ with generator $-a^{3} - a$ with minimal polynomial $x^{2} + 2$:

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

  Sato Tate group: $J(E_1)$
  Of $\GL_2$-type, simple

Over subfield $F \simeq$ $\Q(\sqrt{2})$ with generator $a^{3} - a$ with minimal polynomial $x^{2} - 2$:

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

  Sato Tate group: $J(E_1)$
  Of $\GL_2$-type, simple

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

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

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