# Properties

 Label 1269.b 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 1269.b

Label Equation
1269.b.102789.1 $$y^2 + (x^3 + x)y = -2x^6 - x^5 - 21x^4 - 8x^3 - 80x^2 - 16x - 103$$
1269.b.102789.2 $$y^2 + xy = x^5 - x^4 + x^2 + x$$

## L-function data

Analytic rank:$$0$$

Prime L-Factor
$$3$$$$1 - T$$
$$47$$$$( 1 + T )( 1 - 8 T + 47 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$( 1 - T + 2 T^{2} )( 1 + 2 T + 2 T^{2} )$$
$$5$$$$1 - T - 5 T^{3} + 25 T^{4}$$
$$7$$$$1 + 3 T + 6 T^{2} + 21 T^{3} + 49 T^{4}$$
$$11$$$$1 + 3 T + 2 T^{2} + 33 T^{3} + 121 T^{4}$$
$$13$$$$( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} )$$
$$17$$$$( 1 - 6 T + 17 T^{2} )( 1 + 2 T + 17 T^{2} )$$
$$19$$$$1 - 2 T - 2 T^{2} - 38 T^{3} + 361 T^{4}$$
$$23$$$$1 + 5 T + 30 T^{2} + 115 T^{3} + 529 T^{4}$$
$$29$$$$( 1 - 2 T + 29 T^{2} )( 1 + 5 T + 29 T^{2} )$$
$\cdots$$\cdots$

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