Genus 2 curve 4096.b.65536.1

• Label: 4096.b.65536.1
• Conductor: $4096$
• Discriminant: $-65536$
• Mordell-Weil group: \(\Z/{2}\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
• Sato-Tate group: $J(C_2)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\C)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\mathsf{CM})\)
• \(\End(J) \otimes \Q\): \(\mathsf{CM}\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

Minimal equation

$y^2 = x^5 - x$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(4096\)	\(=\)	\( 2^{12} \)
Discriminant:	\( \Delta \)	\(=\)	\(-65536\)	\(=\)	\( - 2^{16} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(20\)	\(=\)	\( 2^{2} \cdot 5 \)
\( I_4 \)	\(=\)	\(-20\)	\(=\)	\( - 2^{2} \cdot 5 \)
\( I_6 \)	\(=\)	\(-40\)	\(=\)	\( - 2^{3} \cdot 5 \)
\( I_{10} \)	\(=\)	\(8\)	\(=\)	\( 2^{3} \)

Automorphism group

\(\mathrm{Aut}(X)\)	\(\simeq\)	$C_4$
\(\mathrm{Aut}(X_{\overline{\Q}})\)	\(\simeq\)	$GL(2,3)$

Rational points

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

Number of rational Weierstrass points: \(4\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z/{2}\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)

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

2-torsion field: \(\Q(\sqrt{-1}) \)

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(3\)
Regulator:	\( 1 \)
Real period:	\( 12.68998 \)
Tamagawa product:	\( 4 \)
Torsion order:	\( 8 \)
Leading coefficient:	\( 0.793124 \)
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?
\(2\)	\(12\)	\(16\)	\(4\)	\(1^*\)	\(1\)		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?
\(2\)	2.360.2	yes
\(3\)	3.6480.22	no

Sato-Tate group

\(\mathrm{ST}\)	\(\simeq\)	$J(C_2)$
\(\mathrm{ST}^0\)	\(\simeq\)	\(\mathrm{U}(1)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) \(\Q(\sqrt{2}) \) with defining polynomial:
  \(x^{2} - 2\)

Decomposes up to isogeny as the square of the elliptic curve isogeny class:
  Elliptic curve isogeny class 2.2.8.1-64.1-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 \(K = \Q (a) \simeq \) \(\Q(\zeta_{8})\) 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 maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)	\(\simeq\)	\(\mathrm{M}_2(\)\(\Q(\sqrt{-2}) \)\()\)
\(\End (J_{\overline{\Q}}) \otimes \R\)	\(\simeq\)	\(\mathrm{M}_2 (\C)\)

Remainder of the endomorphism lattice by field

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

\(\End (J_{F})\)	\(\simeq\)	the maximal order of \(\End (J_{F}) \otimes \Q\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	\(\Q(\zeta_{8})\) (CM)
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\C \times \C\)

  Sato Tate group: $C_2$
  Not of \(\GL_2\)-type, simple

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

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

  Sato Tate group: $C_{2,1}$
  Not of \(\GL_2\)-type, not simple

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

\(\End (J_{F})\)	\(\simeq\)	a maximal order of \(\End (J_{F}) \otimes \Q\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	the quaternion algebra over \(\Q\) of discriminant 2
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\H\)

  Sato Tate group: $J(C_1)$
  Not of \(\GL_2\)-type, simple