# Properties

 Label 100277.c 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 100277.c

Label Equation
100277.c.100277.1 $$y^2 + y = x^5 - 4x^4 + 4x^2 - 2x$$

## L-function data

Analytic rank:$$1$$

Prime L-Factor
$$149$$$$( 1 + T )( 1 - 15 T + 149 T^{2} )$$
$$673$$$$( 1 - T )( 1 + T + 673 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$( 1 - 2 T + 2 T^{2} )( 1 + 2 T + 2 T^{2} )$$
$$3$$$$1 - 2 T^{2} + 9 T^{4}$$
$$5$$$$( 1 - 2 T + 5 T^{2} )( 1 + 2 T + 5 T^{2} )$$
$$7$$$$( 1 - 2 T + 7 T^{2} )( 1 + 2 T + 7 T^{2} )$$
$$11$$$$1 + T + 11 T^{2} + 11 T^{3} + 121 T^{4}$$
$$13$$$$1 + 3 T + 18 T^{2} + 39 T^{3} + 169 T^{4}$$
$$17$$$$1 - 2 T + 12 T^{2} - 34 T^{3} + 289 T^{4}$$
$$19$$$$1 + 6 T + 40 T^{2} + 114 T^{3} + 361 T^{4}$$
$$23$$$$1 + 5 T + 28 T^{2} + 115 T^{3} + 529 T^{4}$$
$$29$$$$1 - 9 T + 68 T^{2} - 261 T^{3} + 841 T^{4}$$
$\cdots$$\cdots$

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