# Properties

 Label 1647.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 1647.a

Label Equation
1647.a.1647.1 $$y^2 + (x^3 + x + 1)y = x^5$$

## L-function data

Analytic rank:$$1$$
Mordell-Weil rank:$$1$$

Prime L-Factor
$$3$$$$1 + 3 T + 3 T^{2}$$
$$61$$$$( 1 - T )( 1 + 6 T + 61 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$1 + T + 3 T^{2} + 2 T^{3} + 4 T^{4}$$
$$5$$$$( 1 - 3 T + 5 T^{2} )( 1 + 3 T + 5 T^{2} )$$
$$7$$$$1 + 6 T + 20 T^{2} + 42 T^{3} + 49 T^{4}$$
$$11$$$$1 - 2 T + 13 T^{2} - 22 T^{3} + 121 T^{4}$$
$$13$$$$1 + 3 T - T^{2} + 39 T^{3} + 169 T^{4}$$
$$17$$$$1 - T - 13 T^{2} - 17 T^{3} + 289 T^{4}$$
$$19$$$$1 - 17 T^{2} + 361 T^{4}$$
$$23$$$$1 + 8 T + 51 T^{2} + 184 T^{3} + 529 T^{4}$$
$$29$$$$1 - 2 T - 8 T^{2} - 58 T^{3} + 841 T^{4}$$
$\cdots$$\cdots$

## Sato-Tate group

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

## Endomorphisms of the Jacobian

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

Endomorphism algebra over $$\Q$$:

 $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R$$

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.