Properties

Label 324.a
Sato-Tate group $E_3$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\R)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\Q)\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

Related objects

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Genus 2 curves in isogeny class 324.a

Label Equation
324.a.648.1 \(y^2 + (x^3 + x + 1)y = x^5 + 2x^4 + 2x^3 + x^2\)

L-function data

Analytic rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(2\)\( 1 + T + T^{2}\)
\(3\)\( 1 + 3 T + 3 T^{2}\)
 
Good L-factors:
Prime L-Factor
\(5\)\( 1 - 5 T^{2} + 25 T^{4}\)
\(7\)\( 1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4}\)
\(11\)\( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4}\)
\(13\)\( ( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )\)
\(17\)\( ( 1 + 3 T + 17 T^{2} )^{2}\)
\(19\)\( ( 1 + T + 19 T^{2} )^{2}\)
\(23\)\( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4}\)
\(29\)\( 1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4}\)
$\cdots$$\cdots$
 
See L-function page for more information

Sato-Tate group

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

Endomorphisms of the Jacobian

Of \(\GL_2\)-type over \(\Q\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\zeta_{9})^+\) with defining polynomial \(x^{3} - 3 x - 1\)

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.