Properties

Label 405769.a
Conductor $405769$
Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathsf{RM}\)
\(\End(J) \otimes \Q\) \(\mathsf{RM}\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type yes

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Genus 2 curves in isogeny class 405769.a

Label Equation
405769.a.405769.1 \(y^2 + (x^3 + x + 1)y = 2x^5 + 2x^4 + 3x^3 + x^2\)

L-function data

Analytic rank:\(2\)  (upper bound)
Mordell-Weil rank:\(2\)
 
Bad L-factors:
Prime L-Factor
\(7\)\( 1\)
\(13\)\( ( 1 - T )^{2}\)
 
Good L-factors:
Prime L-Factor
\(2\)\( 1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4}\)
\(3\)\( 1 + T^{2} + 9 T^{4}\)
\(5\)\( 1 + 5 T^{2} + 25 T^{4}\)
\(11\)\( ( 1 + 3 T + 11 T^{2} )^{2}\)
\(17\)\( 1 - 6 T + 23 T^{2} - 102 T^{3} + 289 T^{4}\)
\(19\)\( ( 1 + 3 T + 19 T^{2} )^{2}\)
\(23\)\( 1 + 12 T + 77 T^{2} + 276 T^{3} + 529 T^{4}\)
\(29\)\( 1 + 38 T^{2} + 841 T^{4}\)
$\cdots$$\cdots$
 
See L-function page for more information

Sato-Tate group

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

Decomposition of the Jacobian

Simple over \(\overline{\Q}\)

Endomorphisms of the Jacobian

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

Endomorphism algebra over \(\Q\):

\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{5}) \)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R \times \R\)

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.