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: $E_4$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\R)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\Q)\)
• \(\End(J) \otimes \Q\): \(\mathsf{CM}\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

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	Root number*	L-factor	Cluster picture	Tame reduction?
\(2\)	\(8\)	\(9\)	\(2\)	\(1^*\)	\(1 + 2 T + 2 T^{2}\)		no

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