Properties

Label 36864.a
Conductor $36864$
Sato-Tate group $N(\mathrm{SU}(2)\times\mathrm{SU}(2))$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathsf{RM}\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

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Genus 2 curves in isogeny class 36864.a

Label Equation
36864.a.36864.1 \(y^2 = x^5 - x^4 - x^3 + x^2 + x - 1\)

L-function data

Analytic rank:\(0\)
Mordell-Weil rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(2\)\( 1\)
\(3\)\( 1 + T^{2}\)
 
Good L-factors:
Prime L-Factor
\(5\)\( 1 + 2 T^{2} + 25 T^{4}\)
\(7\)\( 1 - 4 T + 10 T^{2} - 28 T^{3} + 49 T^{4}\)
\(11\)\( 1 + 10 T^{2} + 121 T^{4}\)
\(13\)\( 1 + 6 T^{2} + 169 T^{4}\)
\(17\)\( 1 + 4 T + 6 T^{2} + 68 T^{3} + 289 T^{4}\)
\(19\)\( 1 + 10 T^{2} + 361 T^{4}\)
\(23\)\( ( 1 - 4 T + 23 T^{2} )^{2}\)
\(29\)\( 1 - 14 T^{2} + 841 T^{4}\)
$\cdots$$\cdots$
 
See L-function page for more information

Sato-Tate group

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

Decomposition of the Jacobian

Simple over \(\overline{\Q}\)

Endomorphisms of the Jacobian

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

Endomorphism algebra over \(\Q\):

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

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 \(\overline{\Q}\):

\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{2}) \)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\R \times \R\)

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