Properties

 Label 708.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 708.a

Label Equation
708.a.2832.1 $$y^2 + (x^2 + x + 1)y = x^5$$
708.a.19116.1 $$y^2 + (x^3 + 1)y = -x^5 + 4x^2 + 4x - 1$$
708.a.181248.1 $$y^2 + (x^3 + 1)y = -x^6 - 4x^5 + 9x^4 + 48x^3 - 41x^2 - 98x - 36$$

L-function data

Analytic rank:$$0$$

Prime L-Factor
$$2$$$$1 + T^{2}$$
$$3$$$$( 1 + T )( 1 + T + 3 T^{2} )$$
$$59$$$$( 1 + T )( 1 + 59 T^{2} )$$

Good L-factors:
Prime L-Factor
$$5$$$$1 - T - 5 T^{3} + 25 T^{4}$$
$$7$$$$1 + 3 T + 6 T^{2} + 21 T^{3} + 49 T^{4}$$
$$11$$$$1 - 2 T + 2 T^{2} - 22 T^{3} + 121 T^{4}$$
$$13$$$$( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} )$$
$$17$$$$( 1 - 6 T + 17 T^{2} )( 1 + 2 T + 17 T^{2} )$$
$$19$$$$1 + 5 T + 18 T^{2} + 95 T^{3} + 361 T^{4}$$
$$23$$$$1 - 8 T + 42 T^{2} - 184 T^{3} + 529 T^{4}$$
$$29$$$$1 + 5 T + 28 T^{2} + 145 T^{3} + 841 T^{4}$$
$\cdots$$\cdots$

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