# Properties

 Label 1051.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 1051.a

Label Equation
1051.a.1051.1 $$y^2 + y = x^5 - x^4 + x^2 - x$$

## L-function data

Analytic rank:$$1$$

Prime L-Factor
$$1051$$$$( 1 - T )( 1 + 36 T + 1051 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$( 1 + 2 T^{2} )( 1 + 2 T + 2 T^{2} )$$
$$3$$$$( 1 + 3 T^{2} )( 1 + 3 T + 3 T^{2} )$$
$$5$$$$1 + T - T^{2} + 5 T^{3} + 25 T^{4}$$
$$7$$$$( 1 - 2 T + 7 T^{2} )( 1 + 5 T + 7 T^{2} )$$
$$11$$$$1 + T - T^{2} + 11 T^{3} + 121 T^{4}$$
$$13$$$$1 - T + 8 T^{2} - 13 T^{3} + 169 T^{4}$$
$$17$$$$( 1 - 6 T + 17 T^{2} )( 1 + 17 T^{2} )$$
$$19$$$$( 1 + 4 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$
$$23$$$$1 - T + 18 T^{2} - 23 T^{3} + 529 T^{4}$$
$$29$$$$1 + 7 T + 32 T^{2} + 203 T^{3} + 841 T^{4}$$
$\cdots$$\cdots$

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