Properties

 Label 1270.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 1270.a

Label Equation
1270.a.1270.1 $$y^2 + (x^3 + 1)y = -2x^4 + 5x^2 - 2x - 3$$
1270.a.325120.1 $$y^2 + (x^2 + x)y = x^5 + 17x^4 + 76x^3 + 14x^2 - 32x + 3$$

L-function data

Analytic rank:$$0$$

Prime L-Factor
$$2$$$$( 1 + T )( 1 - T + 2 T^{2} )$$
$$5$$$$( 1 - T )( 1 + 3 T + 5 T^{2} )$$
$$127$$$$( 1 - T )( 1 - 20 T + 127 T^{2} )$$

Good L-factors:
Prime L-Factor
$$3$$$$( 1 - 2 T + 3 T^{2} )( 1 + 2 T + 3 T^{2} )$$
$$7$$$$( 1 - 4 T + 7 T^{2} )( 1 + T + 7 T^{2} )$$
$$11$$$$1 + 3 T + 10 T^{2} + 33 T^{3} + 121 T^{4}$$
$$13$$$$( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )$$
$$17$$$$( 1 - 7 T + 17 T^{2} )( 1 + 6 T + 17 T^{2} )$$
$$19$$$$1 - 5 T + 26 T^{2} - 95 T^{3} + 361 T^{4}$$
$$23$$$$( 1 - 4 T + 23 T^{2} )( 1 + 9 T + 23 T^{2} )$$
$$29$$$$( 1 - 6 T + 29 T^{2} )( 1 + 2 T + 29 T^{2} )$$
$\cdots$$\cdots$

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