Properties

Label 11664.a
Sato-Tate group $D_{6,2}$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\C)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\mathrm{CM})\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

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Genus 2 curves in isogeny class 11664.a

Label Equation
11664.a.11664.1 \(y^2 + y = -x^6\)

L-function data

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

Sato-Tate group

\(\mathrm{ST} =\) $D_{6,2}$, \(\quad \mathrm{ST}^0 = \mathrm{U}(1)\)

Endomorphisms of the Jacobian

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

Endomorphism algebra over \(\Q\):

\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R \times \R\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a)\) with defining polynomial \(x^{12} - 3 x^{10} - 8 x^{9} - 6 x^{8} + 12 x^{7} + 47 x^{6} + 78 x^{5} + 78 x^{4} + 50 x^{3} + 21 x^{2} + 6 x + 1\)

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

\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\mathrm{M}_2(\)\(\Q(\sqrt{-3}) \)\()\)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\mathrm{M}_2 (\C)\)

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