Genus 2 curve 256.a.512.1

• Label: 256.a.512.1
• Conductor: $256$
• Discriminant: $-512$
• Mordell-Weil group: $\Z/{2}\Z \oplus \Z/{10}\Z$
• 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

This is a model for the modular curve $X_1(16)$.

Minimal equation

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

(, )

Invariants

Conductor:	$N$	$=$	$256$	$=$	$2^{8}$
Discriminant:	$\Delta$	$=$	$-512$	$=$	$- 2^{9}$

Igusa-Clebsch invariants

$I_2$	$=$	$26$	$=$	$2 \cdot 13$
$I_4$	$=$	$-2$	$=$	$-2$
$I_6$	$=$	$40$	$=$	$2^{3} \cdot 5$
$I_{10}$	$=$	$2$	$=$	$2$

Automorphism group

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

Rational points

All points: $(1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (1 : 0 : 1),\, (1 : -1 : 1),\, (1 : -4 : 2)$

Number of rational Weierstrass points: $2$

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: $\Z/{2}\Z \oplus \Z/{10}\Z$

Generator	$D_0$						Height	Order
$(1 : -4 : 2) - (1 : 0 : 0)$	$2x - z$	$=$	$0,$	$2y$	$=$	$-z^3$	$0$	$2$
$(1 : 0 : 1) - (1 : 0 : 0)$	$x - z$	$=$	$0,$	$y$	$=$	$0$	$0$	$10$

2-torsion field: $\Q(\zeta_{8})$

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	$0$
Mordell-Weil rank:	$0$
2-Selmer rank:	$2$
Regulator:	$1$
Real period:	$26.84182$
Tamagawa product:	$2$
Torsion order:	$20$
Leading coefficient:	$0.134209$
Analytic order of Ш:	$1$   (rounded)
Order of Ш:	square

Local invariants

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

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?
$2$	2.180.3	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 $\Q (b) \simeq$ $\Q(\zeta_{16})^+$ with defining polynomial:
  $x^{4} - 4 x^{2} + 2$

Decomposes up to isogeny as the square of the elliptic curve isogeny class:
  Elliptic curve isogeny class 4.4.2048.1-1.1-a

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