Properties

Label 69696.c
Conductor $69696$
Sato-Tate group $E_2$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\R)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\Q)\)
\(\End(J) \otimes \Q\) \(\mathsf{CM}\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

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Genus 2 curves in isogeny class 69696.c

Label Equation
69696.c.627264.1 \(y^2 + (x^3 + x^2 + x + 1)y = x^6 + 3x^4 - x^3 + 3x^2 - x + 1\)

L-function data

Analytic rank:\(0\)
Mordell-Weil rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(2\)\( 1 - 2 T + 2 T^{2}\)
\(3\)\( 1 + T^{2}\)
\(11\)\( 1 + T^{2}\)
 
Good L-factors:
Prime L-Factor
\(5\)\( ( 1 - 5 T^{2} )^{2}\)
\(7\)\( ( 1 - 4 T + 7 T^{2} )^{2}\)
\(13\)\( 1 - 22 T^{2} + 169 T^{4}\)
\(17\)\( ( 1 - 2 T + 17 T^{2} )^{2}\)
\(19\)\( 1 - 22 T^{2} + 361 T^{4}\)
\(23\)\( ( 1 + 6 T + 23 T^{2} )^{2}\)
\(29\)\( 1 - 22 T^{2} + 841 T^{4}\)
$\cdots$$\cdots$
 
See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $E_2$, \(\quad \mathrm{ST}^0 = \mathrm{SU}(2)\)

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 2.2.8.1-1089.1-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 \(K = \Q (a) \simeq \) \(\Q(\sqrt{2}) \) with defining polynomial \(x^{2} - 2\)

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

\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\mathrm{M}_2 (\R)\)

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