# Properties

 Label 100352.a Sato-Tate group $N(G_{3,3})$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type no

# Related objects

## Genus 2 curves in isogeny class 100352.a

Label Equation
100352.a.100352.1 $$y^2 = x^5 - x^4 + x^2 + x$$

## L-function data

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

Prime L-Factor
$$2$$$$1$$
$$7$$$$1 + T^{2}$$

Good L-factors:
Prime L-Factor
$$3$$$$1 - 2 T^{2} + 9 T^{4}$$
$$5$$$$( 1 - 2 T + 5 T^{2} )( 1 + 5 T^{2} )$$
$$11$$$$( 1 - 4 T + 11 T^{2} )( 1 + 4 T + 11 T^{2} )$$
$$13$$$$( 1 - 6 T + 13 T^{2} )( 1 + 13 T^{2} )$$
$$17$$$$( 1 - 8 T + 17 T^{2} )( 1 + 17 T^{2} )$$
$$19$$$$1 - 10 T^{2} + 361 T^{4}$$
$$23$$$$( 1 - 4 T + 23 T^{2} )( 1 + 4 T + 23 T^{2} )$$
$$29$$$$( 1 - 2 T + 29 T^{2} )( 1 + 6 T + 29 T^{2} )$$
$\cdots$$\cdots$

## Sato-Tate group

$$\mathrm{ST} =$$ $N(G_{3,3})$, $$\quad \mathrm{ST}^0 = \mathrm{SU}(2)\times\mathrm{SU}(2)$$

## 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 $$K = \Q (a) \simeq$$ $$\Q(\sqrt{-1})$$ with defining polynomial $$x^{2} + 1$$

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

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

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