Properties

 Label 523.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 523.a

Label Equation
523.a.523.1 $$y^2 + (x + 1)y = x^5 - x^4 - x^3$$
523.a.523.2 $$y^2 + xy = x^5 - 31x^4 - 110x^3 + 21x^2 - x$$

L-function data

Analytic rank:$$0$$

Prime L-Factor
$$523$$$$( 1 + T )( 1 - 24 T + 523 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$( 1 - T + 2 T^{2} )( 1 + 2 T + 2 T^{2} )$$
$$3$$$$1 + 9 T^{4}$$
$$5$$$$( 1 + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$
$$7$$$$( 1 - 4 T + 7 T^{2} )( 1 + 2 T + 7 T^{2} )$$
$$11$$$$1 - 2 T + 2 T^{2} - 22 T^{3} + 121 T^{4}$$
$$13$$$$( 1 - 2 T + 13 T^{2} )( 1 + T + 13 T^{2} )$$
$$17$$$$( 1 - 5 T + 17 T^{2} )( 1 + 2 T + 17 T^{2} )$$
$$19$$$$1 + T - 2 T^{2} + 19 T^{3} + 361 T^{4}$$
$$23$$$$( 1 + T + 23 T^{2} )( 1 + 8 T + 23 T^{2} )$$
$$29$$$$( 1 - 10 T + 29 T^{2} )( 1 + 9 T + 29 T^{2} )$$
$\cdots$$\cdots$

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