# Properties

 Label 20736.g Sato-Tate group $E_6$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\mathrm{M}_2(\R)$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{M}_2(\Q)$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

# Related objects

## Genus 2 curves in isogeny class 20736.g

Label Equation
20736.g.186624.1 $$y^2 = x^5 + 2x^4 - 9x^3 + 7x^2 - x$$

## L-function data

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

Prime L-Factor
$$2$$$$1$$
$$3$$$$1 + 3 T^{2}$$

Good L-factors:
Prime L-Factor
$$5$$$$1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4}$$
$$7$$$$( 1 - T + 7 T^{2} )( 1 + 4 T + 7 T^{2} )$$
$$11$$$$1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4}$$
$$13$$$$( 1 - 7 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )$$
$$17$$$$1 + 14 T^{2} + 289 T^{4}$$
$$19$$$$( 1 - 8 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$
$$23$$$$1 - 9 T + 58 T^{2} - 207 T^{3} + 529 T^{4}$$
$$29$$$$1 - 3 T + 32 T^{2} - 87 T^{3} + 841 T^{4}$$
$\cdots$$\cdots$

## Sato-Tate group

$$\mathrm{ST} =$$ $E_6$, $$\quad \mathrm{ST}^0 = \mathrm{SU}(2)$$

## Endomorphisms of the Jacobian

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

Endomorphism algebra over $$\Q$$:

 $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{-3})$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\C$$

Smallest field over which all endomorphisms are defined:
Galois number field $$K = \Q (a) \simeq$$ $$\Q(\zeta_{36})^+$$ with defining polynomial $$x^{6} - 6 x^{4} + 9 x^{2} - 3$$

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.