Properties

 Label 691.a Sato-Tate group $\mathrm{USp}(4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

Related objects

Genus 2 curves in isogeny class 691.a

Label Equation
691.a.691.1 $$y^2 + (x + 1)y = x^5 - x^3 - x^2$$

L-function data

Analytic rank:$$0$$

Prime L-Factor
$$691$$$$( 1 + T )( 1 + 28 T + 691 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$1 + T + 2 T^{3} + 4 T^{4}$$
$$3$$$$1 - 2 T^{2} + 9 T^{4}$$
$$5$$$$( 1 + 2 T + 5 T^{2} )^{2}$$
$$7$$$$1 + T + 6 T^{2} + 7 T^{3} + 49 T^{4}$$
$$11$$$$1 - 2 T + 6 T^{2} - 22 T^{3} + 121 T^{4}$$
$$13$$$$( 1 - 6 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )$$
$$17$$$$( 1 - 2 T + 17 T^{2} )( 1 + 2 T + 17 T^{2} )$$
$$19$$$$( 1 - 4 T + 19 T^{2} )( 1 + 2 T + 19 T^{2} )$$
$$23$$$$1 - 10 T^{2} + 529 T^{4}$$
$$29$$$$1 - 3 T + 16 T^{2} - 87 T^{3} + 841 T^{4}$$
$\cdots$$\cdots$

$$\mathrm{ST} =$$ $\mathrm{USp}(4)$
Not of $$\GL_2$$-type over $$\Q$$
All $$\overline{\Q}$$-endomorphisms of the Jacobian are defined over $$\Q$$.