Genus 2 curve 65536.a.65536.1

• Label: 65536.a.65536.1
• Conductor: $65536$
• Discriminant: $65536$
• Mordell-Weil group: \(\Z \oplus \Z/{2}\Z\)
• Sato-Tate group: $D_{2,1}$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\C)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\mathsf{CM})\)
• \(\End(J) \otimes \Q\): \(\Q \times \Q\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

Minimal equation

$y^2 = x^5 + x$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(65536\)	\(=\)	\( 2^{16} \)
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

\(\mathrm{Aut}(X)\)	\(\simeq\)	$C_2^2$
\(\mathrm{Aut}(X_{\overline{\Q}})\)	\(\simeq\)	$GL(2,3)$

Rational points

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

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
\(D_0 - 2 \cdot(1 : 0 : 0)\)	\(x^2 + z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(xz^2 + z^3\)	\(0.960251\)	\(\infty\)
\((0 : 0 : 1) - (1 : 0 : 0)\)	\(x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0\)	\(2\)

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

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(1\)
Mordell-Weil rank:	\(1\)
2-Selmer rank:	\(2\)
Regulator:	\( 0.960251 \)
Real period:	\( 8.973175 \)
Tamagawa product:	\( 1 \)
Torsion order:	\( 2 \)
Leading coefficient:	\( 2.154126 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(16\)	\(16\)	\(1\)	\(1\)

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?
\(2\)	2.180.5	yes
\(3\)	3.6480.21	no

Sato-Tate group

\(\mathrm{ST}\)	\(\simeq\)	$D_{2,1}$
\(\mathrm{ST}^0\)	\(\simeq\)	\(\mathrm{U}(1)\)

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  Elliptic curve isogeny class 256.d
  Elliptic curve isogeny class 256.a

Endomorphisms of the Jacobian

Of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)	\(\simeq\)	an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q\) \(\times\) \(\Q\)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\R \times \R\)

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 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 \(x^{2} + 2\):

\(\End (J_{F})\)	\(\simeq\)	an order of index \(4\) in \(\Z [\sqrt{-2}] \times \Z [\sqrt{-2}]\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{-2}) \) \(\times\) \(\Q(\sqrt{-2}) \)
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\C \times \C\)

  Sato Tate group: $C_2$
  Not of \(\GL_2\)-type, not 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 \(8\) 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 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