Properties

 Label 249.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 249.a

Label Equation
249.a.249.1 $$y^2 + (x^3 + 1)y = x^2 + x$$
249.a.6723.1 $$y^2 + (x^3 + 1)y = -x^5 + x^3 + x^2 + 3x + 2$$

L-function data

Analytic rank:$$0$$

Prime L-Factor
$$3$$$$( 1 - T )( 1 + 3 T + 3 T^{2} )$$
$$83$$$$( 1 - T )( 1 + 83 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$1 + 2 T + 3 T^{2} + 4 T^{3} + 4 T^{4}$$
$$5$$$$1 + 2 T^{2} + 25 T^{4}$$
$$7$$$$1 + T - 2 T^{2} + 7 T^{3} + 49 T^{4}$$
$$11$$$$( 1 - 5 T + 11 T^{2} )( 1 + 4 T + 11 T^{2} )$$
$$13$$$$1 - 2 T^{2} + 169 T^{4}$$
$$17$$$$( 1 - 2 T + 17 T^{2} )( 1 + 3 T + 17 T^{2} )$$
$$19$$$$( 1 - 2 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$
$$23$$$$1 - 26 T^{2} + 529 T^{4}$$
$$29$$$$1 - 3 T + 32 T^{2} - 87 T^{3} + 841 T^{4}$$
$\cdots$$\cdots$

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