# Properties

 Label 1147.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 1147.a

Label Equation
1147.a.35557.1 $$y^2 + xy = x^5 + 8x^4 + 18x^3 + 8x^2 + x$$
1147.a.35557.2 $$y^2 + xy = x^5 + 6x^4 - 32x^2 + x$$

## L-function data

Analytic rank:$$0$$

Prime L-Factor
$$31$$$$( 1 + T )( 1 + 31 T^{2} )$$
$$37$$$$( 1 + T )( 1 - 6 T + 37 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$1 + T + 2 T^{3} + 4 T^{4}$$
$$3$$$$1 + T + 2 T^{2} + 3 T^{3} + 9 T^{4}$$
$$5$$$$( 1 - 2 T + 5 T^{2} )( 1 + 2 T + 5 T^{2} )$$
$$7$$$$1 - 3 T + 6 T^{2} - 21 T^{3} + 49 T^{4}$$
$$11$$$$1 + 3 T + 2 T^{2} + 33 T^{3} + 121 T^{4}$$
$$13$$$$( 1 - 2 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )$$
$$17$$$$( 1 - 2 T + 17 T^{2} )( 1 + 4 T + 17 T^{2} )$$
$$19$$$$1 - 2 T - 2 T^{2} - 38 T^{3} + 361 T^{4}$$
$$23$$$$( 1 - 8 T + 23 T^{2} )( 1 + 8 T + 23 T^{2} )$$
$$29$$$$1 - 2 T - 14 T^{2} - 58 T^{3} + 841 T^{4}$$
$\cdots$$\cdots$

## Sato-Tate group

$$\mathrm{ST} =$$ $\mathrm{USp}(4)$

## Endomorphisms of the Jacobian

Not of $$\GL_2$$-type over $$\Q$$

All $$\overline{\Q}$$-endomorphisms of the Jacobian are defined over $$\Q$$.

More complete information on endomorphism algebras and rings can be found on the pages of the individual curves in the isogeny class.