# Properties

 Label 1148.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 1148.a

Label Equation
1148.a.8036.1 $$y^2 + (x^3 + 1)y = x^5 - x^4 - 6x^3 + x^2 + 5x - 1$$
1148.a.47068.1 $$y^2 + (x^2 + x + 1)y = x^5 + 2x^4 - 5x^3 + x$$

## L-function data

Analytic rank:$$0$$

Prime L-Factor
$$2$$$$1 + T^{2}$$
$$7$$$$( 1 - T )( 1 + 2 T + 7 T^{2} )$$
$$41$$$$( 1 - T )( 1 + 8 T + 41 T^{2} )$$

Good L-factors:
Prime L-Factor
$$3$$$$1 + 9 T^{4}$$
$$5$$$$( 1 - 2 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$
$$11$$$$1 - 2 T + 2 T^{2} - 22 T^{3} + 121 T^{4}$$
$$13$$$$1 + 4 T + 24 T^{2} + 52 T^{3} + 169 T^{4}$$
$$17$$$$1 - 10 T^{2} + 289 T^{4}$$
$$19$$$$1 + 4 T + 28 T^{2} + 76 T^{3} + 361 T^{4}$$
$$23$$$$( 1 - 4 T + 23 T^{2} )^{2}$$
$$29$$$$1 - 2 T - 2 T^{2} - 58 T^{3} + 841 T^{4}$$
$\cdots$$\cdots$

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