# Properties

 Label 3969.b 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 3969.b

Label Equation
3969.b.35721.1 $$y^2 + (x^3 + x + 1)y = -2x^5 + 3x^4 - 3x^2$$

## L-function data

Analytic rank:$$2$$

Prime L-Factor
$$3$$$$1 + 3 T + 3 T^{2}$$
$$7$$$$1 + 5 T + 7 T^{2}$$

Good L-factors:
Prime L-Factor
$$2$$$$1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4}$$
$$5$$$$( 1 + 3 T + 5 T^{2} )^{2}$$
$$11$$$$1 - 19 T^{2} + 121 T^{4}$$
$$13$$$$( 1 - 2 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} )$$
$$17$$$$1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4}$$
$$19$$$$( 1 + T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$
$$23$$$$1 - 19 T^{2} + 529 T^{4}$$
$$29$$$$1 + 9 T + 56 T^{2} + 261 T^{3} + 841 T^{4}$$
$\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 $$\Q$$

Smallest field over which all endomorphisms are defined:
Galois number field $$K = \Q (a) \simeq$$ 6.6.330812181.1 with defining polynomial $$x^{6} - 21 x^{4} - 14 x^{3} + 63 x^{2} + 21 x - 35$$

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.