# Properties

 Label 3721.a 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

# Learn more about

## Genus 2 curves in isogeny class 3721.a

Label Equation
3721.a.3721.1 $$y^2 + (x^3 + x + 1)y = -x^4 + x^3 + 3x^2 + x$$

## L-function data

Analytic rank:$$2$$

Bad L-factors:
Prime L-Factor
$$61$$$$1 + T + 61 T^{2}$$

Good L-factors:
Prime L-Factor
$$2$$$$1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4}$$
$$3$$$$( 1 + 2 T + 3 T^{2} )^{2}$$
$$5$$$$1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4}$$
$$7$$$$( 1 + T + 7 T^{2} )( 1 + 5 T + 7 T^{2} )$$
$$11$$$$1 - 10 T^{2} + 121 T^{4}$$
$$13$$$$( 1 - 7 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} )$$
$$17$$$$1 + 12 T + 65 T^{2} + 204 T^{3} + 289 T^{4}$$
$$19$$$$1 - 2 T - 15 T^{2} - 38 T^{3} + 361 T^{4}$$
$$23$$$$( 1 - 23 T^{2} )^{2}$$
$$29$$$$1 + 3 T + 32 T^{2} + 87 T^{3} + 841 T^{4}$$
$\cdots$$\cdots$

See L-function page for more information

## 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 $$K = \Q (a) \simeq$$ 6.6.844596301.1 with defining polynomial $$x^{6} - x^{5} - 25 x^{4} - 8 x^{3} + 123 x^{2} + 126 x + 27$$

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.