# Properties

 Label 256.a Sato-Tate group $E_4$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\mathrm{M}_2(\R)$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{M}_2(\Q)$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

## Genus 2 curves in isogeny class 256.a

Label Equation
256.a.512.1 $$y^2 + y = 2x^5 - 3x^4 + x^3 + x^2 - x$$

## L-function data

Analytic rank:$$0$$

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

Good L-factors:
Prime L-Factor
$$3$$$$1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4}$$
$$5$$$$( 1 - 2 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$
$$7$$$$1 - 10 T^{2} + 49 T^{4}$$
$$11$$$$1 - 2 T + 2 T^{2} - 22 T^{3} + 121 T^{4}$$
$$13$$$$( 1 - 4 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} )$$
$$17$$$$( 1 + 2 T + 17 T^{2} )^{2}$$
$$19$$$$1 - 6 T + 18 T^{2} - 114 T^{3} + 361 T^{4}$$
$$23$$$$1 - 10 T^{2} + 529 T^{4}$$
$$29$$$$( 1 - 10 T + 29 T^{2} )( 1 + 4 T + 29 T^{2} )$$
$\cdots$$\cdots$

## Sato-Tate group

$$\mathrm{ST} =$$ $E_4$, $$\quad \mathrm{ST}^0 = \mathrm{SU}(2)$$

## Endomorphisms of the Jacobian

Of $$\GL_2$$-type over $$\Q$$

Smallest field over which all endomorphisms are defined:
Galois number field $$K = \Q (a) \simeq$$ $$\Q(\zeta_{16})^+$$ with defining polynomial $$x^{4} - 4 x^{2} + 2$$

Endomorphism algebra over $$\overline{\Q}$$:
 $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\simeq$$ $$\mathrm{M}_2($$$$\Q$$$$)$$ $$\End (J_{\overline{\Q}}) \otimes \R$$ $$\simeq$$ $$\mathrm{M}_2 (\R)$$

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