# Properties

 Label 1205.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 1205.a

Label Equation
1205.a.1205.1 $$y^2 + y = x^5 + 2x^4 - x^2$$

## L-function data

Analytic rank:$$1$$

Prime L-Factor
$$5$$$$( 1 + T )( 1 + 3 T + 5 T^{2} )$$
$$241$$$$( 1 - T )( 1 + 24 T + 241 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$( 1 + 2 T^{2} )( 1 + 2 T + 2 T^{2} )$$
$$3$$$$1 + 2 T + 4 T^{2} + 6 T^{3} + 9 T^{4}$$
$$7$$$$1 - T - 3 T^{2} - 7 T^{3} + 49 T^{4}$$
$$11$$$$( 1 - 3 T + 11 T^{2} )( 1 + 6 T + 11 T^{2} )$$
$$13$$$$1 - T + 13 T^{2} - 13 T^{3} + 169 T^{4}$$
$$17$$$$1 - T - 20 T^{2} - 17 T^{3} + 289 T^{4}$$
$$19$$$$1 + 4 T + 12 T^{2} + 76 T^{3} + 361 T^{4}$$
$$23$$$$1 - 3 T + 15 T^{2} - 69 T^{3} + 529 T^{4}$$
$$29$$$$1 + 11 T + 77 T^{2} + 319 T^{3} + 841 T^{4}$$
$\cdots$$\cdots$

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