Properties

Label 961.a
Sato-Tate group $G_{3,3}$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{RM}\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type yes

Related objects

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Genus 2 curves in isogeny class 961.a

Label Equation
961.a.961.1 \(y^2 + (x^3 + x + 1)y = -x^6 - x^5 - 7x^4 + 74x^3 - 145x^2 + 99x - 33\)
961.a.961.2 \(y^2 + (x^3 + x + 1)y = -x^6 + 2x^5 - 8x^4 + 12x^3 - 18x^2 + 12x - 7\)
961.a.961.3 \(y^2 + (x^3 + x + 1)y = x^5 + x^4 + x^3 - x - 1\)
961.a.923521.1 \(y^2 + (x^3 + x^2 + 1)y = -5x^4 + 4x^3 + 3x^2 - 2x - 3\)

L-function data

Analytic rank:\(0\)
 
Bad L-factors:
Prime L-Factor
\(31\)\( ( 1 - T )^{2}\)
 
Good L-factors:
Prime L-Factor
\(2\)\( 1 - T + 3 T^{2} - 2 T^{3} + 4 T^{4}\)
\(3\)\( 1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4}\)
\(5\)\( ( 1 - T + 5 T^{2} )^{2}\)
\(7\)\( 1 + 4 T + 13 T^{2} + 28 T^{3} + 49 T^{4}\)
\(11\)\( ( 1 - 2 T + 11 T^{2} )^{2}\)
\(13\)\( 1 + 2 T + 22 T^{2} + 26 T^{3} + 169 T^{4}\)
\(17\)\( 1 - 6 T + 38 T^{2} - 102 T^{3} + 289 T^{4}\)
\(19\)\( 1 + 33 T^{2} + 361 T^{4}\)
\(23\)\( 1 + 2 T + 2 T^{2} + 46 T^{3} + 529 T^{4}\)
\(29\)\( 1 - 10 T + 78 T^{2} - 290 T^{3} + 841 T^{4}\)
$\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 \(\Q\)

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.