Properties

Label 4096.b
Conductor $4096$
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

Related objects

Learn more

Genus 2 curves in isogeny class 4096.b

Label Equation
4096.b.65536.1 \(y^2 = x^5 - x\)

L-function data

Analytic rank:\(0\)
Mordell-Weil rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(2\)\( 1\)
 
Good L-factors:
Prime L-Factor
\(3\)\( 1 - 2 T^{2} + 9 T^{4}\)
\(5\)\( ( 1 - 5 T^{2} )^{2}\)
\(7\)\( ( 1 + 7 T^{2} )^{2}\)
\(11\)\( 1 + 14 T^{2} + 121 T^{4}\)
\(13\)\( ( 1 - 13 T^{2} )^{2}\)
\(17\)\( ( 1 + 6 T + 17 T^{2} )^{2}\)
\(19\)\( 1 - 34 T^{2} + 361 T^{4}\)
\(23\)\( ( 1 + 23 T^{2} )^{2}\)
\(29\)\( ( 1 - 29 T^{2} )^{2}\)
$\cdots$$\cdots$
 
See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $J(C_2)$, \(\quad \mathrm{ST}^0 = \mathrm{U}(1)\)

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 algebra over \(\Q\):

\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{-1}) \)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\C\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\zeta_{8})\) with defining polynomial \(x^{4} + 1\)

Endomorphism algebra over \(\overline{\Q}\):

\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\mathrm{M}_2(\)\(\Q(\sqrt{-2}) \)\()\)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\mathrm{M}_2 (\C)\)

More complete information on endomorphism algebras and rings can be found on the pages of the individual curves in the isogeny class.