# Properties

 Label 3125.a Sato-Tate group $F_{ac}$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\C \times \C$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{CM}$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

# Related objects

## Genus 2 curves in isogeny class 3125.a

Label Equation
3125.a.3125.1 $$y^2 + y = x^5$$

## L-function data

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

Prime L-Factor
$$5$$$$1$$

Good L-factors:
Prime L-Factor
$$2$$$$( 1 - 2 T + 2 T^{2} )( 1 + 2 T + 2 T^{2} )$$
$$3$$$$1 + 9 T^{4}$$
$$7$$$$1 + 49 T^{4}$$
$$11$$$$1 + T - 9 T^{2} + 11 T^{3} + 121 T^{4}$$
$$13$$$$1 + 169 T^{4}$$
$$17$$$$1 + 289 T^{4}$$
$$19$$$$( 1 + 19 T^{2} )^{2}$$
$$23$$$$1 + 529 T^{4}$$
$$29$$$$( 1 + 29 T^{2} )^{2}$$
$\cdots$$\cdots$

## Sato-Tate group

$$\mathrm{ST} =$$ $F_{ac}$, $$\quad \mathrm{ST}^0 = \mathrm{U}(1)\times\mathrm{U}(1)$$

## Endomorphisms of the Jacobian

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

Endomorphism algebra over $$\Q$$:

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

Smallest field over which all endomorphisms are defined:
Galois number field $$K = \Q (a) \simeq$$ $$\Q(\zeta_{5})$$ with defining polynomial $$x^{4} - x^{3} + x^{2} - x + 1$$

Endomorphism algebra over $$\overline{\Q}$$:

 $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\simeq$$ $$\Q(\zeta_{5})$$ (CM) $$\End (J_{\overline{\Q}}) \otimes \R$$ $$\simeq$$ $$\C \times \C$$

More complete information on endomorphism algebras and rings can be found on the pages of the individual curves in the isogeny class.