Properties

Label 20736.l
Conductor $20736$
Sato-Tate group $J(E_4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\R)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathsf{QM}\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

Related objects

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Genus 2 curves in isogeny class 20736.l

Label Equation
20736.l.373248.1 \(y^2 + y = 6x^5 + 9x^4 - x^3 - 3x^2\)

L-function data

Analytic rank:\(0\)
Mordell-Weil rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(2\)\( 1 + 2 T^{2}\)
\(3\)\( 1\)
 
Good L-factors:
Prime L-Factor
\(5\)\( 1 - 2 T^{2} + 25 T^{4}\)
\(7\)\( 1 - 2 T + 2 T^{2} - 14 T^{3} + 49 T^{4}\)
\(11\)\( 1 - 2 T^{2} + 121 T^{4}\)
\(13\)\( ( 1 - 6 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )\)
\(17\)\( 1 + 10 T^{2} + 289 T^{4}\)
\(19\)\( 1 - 2 T^{2} + 361 T^{4}\)
\(23\)\( 1 - 2 T^{2} + 529 T^{4}\)
\(29\)\( 1 + 46 T^{2} + 841 T^{4}\)
$\cdots$$\cdots$
 
See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $J(E_4)$, \(\quad \mathrm{ST}^0 = \mathrm{SU}(2)\)

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 \(K = \Q (a) \simeq \) 8.0.339738624.10 with defining polynomial \(x^{8} + 4 x^{6} + 10 x^{4} + 24 x^{2} + 36\)

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

\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)the quaternion algebra over \(\Q\) of discriminant 6
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\mathrm{M}_2 (\R)\)

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