Genus 2 curve 135424.l.270848.1

• Label: 135424.l.270848.1
• Conductor: $135424$
• Discriminant: $-270848$
• Mordell-Weil group: $\Z/{2}\Z$
• Sato-Tate group: $E_4$
• 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: $\mathsf{CM}$
• Q‾\overline{\Q}-simple: no
• GL2\mathrm{GL}_2-type: yes

Minimal equation

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

(, )

Invariants

Conductor:	$N$	$=$	$135424$	$=$	$2^{8} \cdot 23^{2}$
Discriminant:	$\Delta$	$=$	$-270848$	$=$	$- 2^{9} \cdot 23^{2}$

Igusa-Clebsch invariants

$I_2$	$=$	$170$	$=$	$2 \cdot 5 \cdot 17$
$I_4$	$=$	$430$	$=$	$2 \cdot 5 \cdot 43$
$I_6$	$=$	$23560$	$=$	$2^{3} \cdot 5 \cdot 19 \cdot 31$
$I_{10}$	$=$	$1058$	$=$	$2 \cdot 23^{2}$

Automorphism group

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

Rational points

This curve has no rational points.

Number of rational Weierstrass points: $0$

This curve is locally solvable except over $\R$.

Mordell-Weil group of the Jacobian

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

Generator	$D_0$						Height	Order
$D_0 - D_\infty$	$x^2 + z^2$	$=$	$0,$	$y$	$=$	$0$	$0$	$2$

2-torsion field: 8.0.34668544.1

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	$0$
Mordell-Weil rank:	$0$
2-Selmer rank:	$2$
Regulator:	$1$
Real period:	$5.075262$
Tamagawa product:	$2$
Torsion order:	$2$
Leading coefficient:	$5.075262$
Analytic order of Ш:	$2$   (rounded)
Order of Ш:	twice a square

Local invariants

Prime	ord($N$)	ord($\Delta$)	Tamagawa	L-factor	Cluster picture
$2$	$8$	$9$	$2$	$1 - 2 T + 2 T^{2}$
$23$	$2$	$2$	$1$	$1 + T^{2}$

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?
$2$	2.45.1	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 $\Q (b) \simeq$ $\Q(\zeta_{16})^+$ with defining polynomial:
  $x^{4} - 4 x^{2} + 2$

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
  $g_4 = 1600 b^{3} - 2880 b^{2} - 1280 b + 2080$
  $g_6 = -484096 b^{3} + 897536 b^{2} + 274688 b - 517632$
   Conductor norm: 279841

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 $K = \Q (a) \simeq$ $\Q(\zeta_{16})^+$ with defining polynomial $x^{4} - 4 x^{2} + 2$

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 $a^{2} - 2$ with minimal polynomial $x^{2} - 2$:

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

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