Properties

Label 1696.b
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 1696.b

Label Equation
1696.b.434176.1 \(y^2 + xy = x^6 - 2x^5 + 2x^4 + 9x^3 - 12x^2 + 3x + 26\)

L-function data

Analytic rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(2\)\( 1 - T\)
\(53\)\( ( 1 + T )( 1 + 53 T^{2} )\)
 
Good L-factors:
Prime L-Factor
\(3\)\( 1 - T^{2} + 9 T^{4}\)
\(5\)\( ( 1 + 3 T + 5 T^{2} )^{2}\)
\(7\)\( 1 - 5 T^{2} + 49 T^{4}\)
\(11\)\( ( 1 - 6 T + 11 T^{2} )( 1 + 6 T + 11 T^{2} )\)
\(13\)\( ( 1 - 5 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )\)
\(17\)\( ( 1 + 17 T^{2} )^{2}\)
\(19\)\( 1 + 7 T^{2} + 361 T^{4}\)
\(23\)\( ( 1 - 6 T + 23 T^{2} )( 1 + 6 T + 23 T^{2} )\)
\(29\)\( ( 1 - 3 T + 29 T^{2} )( 1 + 6 T + 29 T^{2} )\)
$\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 \(K = \Q (a) \simeq \) \(\Q(\sqrt{-1}) \) with defining polynomial \(x^{2} + 1\)

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.