# Properties

 Label 688.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 688.a

Label Equation
688.a.2752.1 $$y^2 + y = 2x^5 - 5x^4 + 4x^3 - x$$
688.a.704512.2 $$y^2 = 2x^5 - 7x^4 - 8x^3 + 2x^2 + 4x + 1$$
688.a.704512.1 $$y^2 = 2x^5 + 4x^3 + x^2 + 2x + 1$$

## L-function data

Analytic rank:$$0$$
Mordell-Weil rank:$$0$$

Prime L-Factor
$$2$$$$1$$
$$43$$$$( 1 - T )( 1 - 4 T + 43 T^{2} )$$

Good L-factors:
Prime L-Factor
$$3$$$$1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4}$$
$$5$$$$( 1 - 2 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$
$$7$$$$( 1 - 2 T + 7 T^{2} )( 1 + 2 T + 7 T^{2} )$$
$$11$$$$1 + 3 T + 2 T^{2} + 33 T^{3} + 121 T^{4}$$
$$13$$$$1 - T + 4 T^{2} - 13 T^{3} + 169 T^{4}$$
$$17$$$$1 - 3 T + 4 T^{2} - 51 T^{3} + 289 T^{4}$$
$$19$$$$1 - 2 T - 2 T^{2} - 38 T^{3} + 361 T^{4}$$
$$23$$$$1 - 5 T + 30 T^{2} - 115 T^{3} + 529 T^{4}$$
$$29$$$$1 + 4 T + 38 T^{2} + 116 T^{3} + 841 T^{4}$$
$\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.