# Properties

 Label 1472.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 1472.a

Label Equation
1472.a.5888.1 $$y^2 = x^5 + x^4 - x^3 - 2x^2 - x$$
1472.a.94208.1 $$y^2 = 4x^5 - 3x^4 - 4x^3 - x^2 + 7x - 3$$
1472.a.94208.2 $$y^2 + y = 4x^5 + x^4 + 4x^2 + 2x$$

## L-function data

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

Prime L-Factor
$$2$$$$1$$
$$23$$$$( 1 + T )( 1 - 8 T + 23 T^{2} )$$

Good L-factors:
Prime L-Factor
$$3$$$$1 + T + 2 T^{2} + 3 T^{3} + 9 T^{4}$$
$$5$$$$( 1 - 2 T + 5 T^{2} )( 1 + 2 T + 5 T^{2} )$$
$$7$$$$( 1 - 4 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} )$$
$$11$$$$( 1 - 4 T + 11 T^{2} )( 1 + 6 T + 11 T^{2} )$$
$$13$$$$1 + 3 T + 12 T^{2} + 39 T^{3} + 169 T^{4}$$
$$17$$$$1 - 2 T + 2 T^{2} - 34 T^{3} + 289 T^{4}$$
$$19$$$$( 1 - 4 T + 19 T^{2} )( 1 + 6 T + 19 T^{2} )$$
$$29$$$$1 - 3 T - 16 T^{2} - 87 T^{3} + 841 T^{4}$$
$\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.