# Properties

 Label 784.c Sato-Tate group $E_3$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\mathrm{M}_2(\R)$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{M}_2(\Q)$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

## Genus 2 curves in isogeny class 784.c

Label Equation
784.c.614656.1 $$y^2 = x^5 - 4x^4 - 13x^3 - 9x^2 - x$$

## L-function data

Analytic rank:$$0$$

Prime L-Factor
$$2$$$$1$$
$$7$$$$1 + 4 T + 7 T^{2}$$

Good L-factors:
Prime L-Factor
$$3$$$$1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4}$$
$$5$$$$1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4}$$
$$11$$$$1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4}$$
$$13$$$$( 1 - 2 T + 13 T^{2} )^{2}$$
$$17$$$$1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4}$$
$$19$$$$( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} )$$
$$23$$$$1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4}$$
$$29$$$$( 1 + 6 T + 29 T^{2} )^{2}$$
$\cdots$$\cdots$

## Sato-Tate group

$$\mathrm{ST} =$$ $E_3$, $$\quad \mathrm{ST}^0 = \mathrm{SU}(2)$$

## Endomorphisms of the Jacobian

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

Smallest field over which all endomorphisms are defined:
Galois number field $$K = \Q (a) \simeq$$ $$\Q(\zeta_{7})^+$$ with defining polynomial $$x^{3} - x^{2} - 2 x + 1$$

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.