Properties

Label 21609.a
Conductor $21609$
Sato-Tate group $G_{3,3}$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathsf{RM}\)
\(\End(J) \otimes \Q\) \(\mathsf{RM}\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type yes

Related objects

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Genus 2 curves in isogeny class 21609.a

Label Equation
21609.a.453789.1 \(y^2 + (x^3 + 1)y = -x^4 + 2x^2 - 3x + 2\)

L-function data

Analytic rank:\(2\)  (upper bound)
Mordell-Weil rank:\(2\)
 
Bad L-factors:
Prime L-Factor
\(3\)\( ( 1 + T )^{2}\)
\(7\)\( 1\)
 
Good L-factors:
Prime L-Factor
\(2\)\( 1 + 2 T + 3 T^{2} + 4 T^{3} + 4 T^{4}\)
\(5\)\( 1 + 4 T + 12 T^{2} + 20 T^{3} + 25 T^{4}\)
\(11\)\( ( 1 + 2 T + 11 T^{2} )^{2}\)
\(13\)\( 1 + 8 T + 40 T^{2} + 104 T^{3} + 169 T^{4}\)
\(17\)\( 1 + 4 T + 20 T^{2} + 68 T^{3} + 289 T^{4}\)
\(19\)\( 1 + 30 T^{2} + 361 T^{4}\)
\(23\)\( 1 + 4 T + 18 T^{2} + 92 T^{3} + 529 T^{4}\)
\(29\)\( 1 + 8 T + 66 T^{2} + 232 T^{3} + 841 T^{4}\)
$\cdots$$\cdots$
 
See L-function page for more information

Sato-Tate group

\(\mathrm{ST} =\) $G_{3,3}$, \(\quad \mathrm{ST}^0 = \mathrm{SU}(2)\times\mathrm{SU}(2)\)

Endomorphisms of the Jacobian

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

Endomorphism algebra over \(\Q\):

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

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).

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