Properties

Label 18225.a
Conductor $18225$
Sato-Tate group $J(E_6)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\R)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\Q)\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type no

Related objects

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Genus 2 curves in isogeny class 18225.a

Label Equation
18225.a.18225.1 \(y^2 + (x^3 + 1)y = 1\)

L-function data

Analytic rank:\(1\)
Mordell-Weil rank:\(1\)
 
Bad L-factors:
Prime L-Factor
\(3\)\( 1\)
\(5\)\( 1 + 5 T^{2}\)
 
Good L-factors:
Prime L-Factor
\(2\)\( 1 + T^{2} + 4 T^{4}\)
\(7\)\( ( 1 + T + 7 T^{2} )( 1 + 5 T + 7 T^{2} )\)
\(11\)\( ( 1 - 3 T + 11 T^{2} )( 1 + 3 T + 11 T^{2} )\)
\(13\)\( ( 1 - 5 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} )\)
\(17\)\( 1 + 22 T^{2} + 289 T^{4}\)
\(19\)\( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4}\)
\(23\)\( 1 - 29 T^{2} + 529 T^{4}\)
\(29\)\( ( 1 - 6 T + 29 T^{2} )( 1 + 6 T + 29 T^{2} )\)
$\cdots$$\cdots$
 
See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $J(E_6)$, \(\quad \mathrm{ST}^0 = \mathrm{SU}(2)\)

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 \(K = \Q (a)\) with defining polynomial \(x^{12} - 6 x^{11} + 27 x^{10} - 80 x^{9} + 195 x^{8} - 366 x^{7} + 551 x^{6} - 642 x^{5} + 585 x^{4} - 400 x^{3} + 177 x^{2} - 42 x + 4\)

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.