# Properties

 Label 6400.f 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 6400.f

Label Equation
6400.f.64000.1 $$y^2 + x^3y = -2x^4 - 3x^3 + x^2 + 6x + 4$$

## L-function data

Analytic rank:$$2$$

Prime L-Factor
$$2$$$$1 + 2 T + 2 T^{2}$$
$$5$$$$1 + 4 T + 5 T^{2}$$

Good L-factors:
Prime L-Factor
$$3$$$$( 1 + 2 T + 3 T^{2} )^{2}$$
$$7$$$$1 + 6 T + 18 T^{2} + 42 T^{3} + 49 T^{4}$$
$$11$$$$1 + 2 T + 2 T^{2} + 22 T^{3} + 121 T^{4}$$
$$13$$$$1 - 22 T^{2} + 169 T^{4}$$
$$17$$$$1 - 2 T + 2 T^{2} - 34 T^{3} + 289 T^{4}$$
$$19$$$$1 + 6 T + 18 T^{2} + 114 T^{3} + 361 T^{4}$$
$$23$$$$1 + 2 T + 2 T^{2} + 46 T^{3} + 529 T^{4}$$
$$29$$$$( 1 + 4 T + 29 T^{2} )( 1 + 10 T + 29 T^{2} )$$
$\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 $$K = \Q (a) \simeq$$ 4.4.256000.1 with defining polynomial $$x^{4} - 20 x^{2} + 10$$

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.