Properties

Label 26244.d
Conductor $26244$
Sato-Tate group $J(E_3)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\R)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\Q)\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type no

Related objects

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Genus 2 curves in isogeny class 26244.d

Label Equation
26244.d.314928.1 \(y^2 + y = x^6 - 2x^3\)

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
\(5\)\( ( 1 - 3 T + 5 T^{2} )( 1 + 3 T + 5 T^{2} )\)
\(7\)\( 1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4}\)
\(11\)\( ( 1 - 6 T + 11 T^{2} )( 1 + 6 T + 11 T^{2} )\)
\(13\)\( ( 1 - 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )\)
\(17\)\( ( 1 - 3 T + 17 T^{2} )( 1 + 3 T + 17 T^{2} )\)
\(19\)\( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4}\)
\(23\)\( ( 1 - 6 T + 23 T^{2} )( 1 + 6 T + 23 T^{2} )\)
\(29\)\( ( 1 - 3 T + 29 T^{2} )( 1 + 3 T + 29 T^{2} )\)
$\cdots$$\cdots$
 
See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $J(E_3)$, \(\quad \mathrm{ST}^0 = \mathrm{SU}(2)\)

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 \(K = \Q (a) \simeq \) 6.0.34992.1 with defining polynomial \(x^{6} - 3 x^{5} + 5 x^{3} - 3 x + 1\)

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

\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\mathrm{M}_2 (\R)\)

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