Properties

Label 3264.a
Sato-Tate group $N(G_{3,3})$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type no

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Genus 2 curves in isogeny class 3264.a

Label Equation
3264.a.3264.1 \(y^2 + x^3y = x^5 - 3x^3 + 4x - 2\)

L-function data

Analytic rank:\(1\)
 
Bad L-factors:
Prime L-Factor
\(2\)\( 1 + 2 T + 2 T^{2}\)
\(3\)\( ( 1 + T )( 1 + 2 T + 3 T^{2} )\)
\(17\)\( ( 1 + T )( 1 + 6 T + 17 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(5\)\( 1 + 2 T^{2} + 25 T^{4}\)
\(7\)\( 1 + 2 T^{2} + 49 T^{4}\)
\(11\)\( ( 1 + 2 T + 11 T^{2} )( 1 + 4 T + 11 T^{2} )\)
\(13\)\( 1 - 6 T^{2} + 169 T^{4}\)
\(19\)\( ( 1 + 19 T^{2} )( 1 + 4 T + 19 T^{2} )\)
\(23\)\( 1 + 6 T^{2} + 529 T^{4}\)
\(29\)\( 1 - 46 T^{2} + 841 T^{4}\)
$\cdots$$\cdots$
 
See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $N(G_{3,3})$, \(\quad \mathrm{ST}^0 = \mathrm{SU}(2)\times\mathrm{SU}(2)\)

Endomorphisms of the Jacobian

Not of \(\GL_2\)-type over \(\Q\)

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

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