# Properties

 Label 21609.a Conductor $21609$ Sato-Tate group $G_{3,3}$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathsf{RM}$$ $$\End(J) \otimes \Q$$ $$\mathsf{RM}$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type yes

# Related objects

## Genus 2 curves in isogeny class 21609.a

Label Equation
21609.a.453789.1 $$y^2 + (x^3 + 1)y = -x^4 + 2x^2 - 3x + 2$$

## L-function data

Analytic rank:$$2$$  (upper bound)
Mordell-Weil rank:$$2$$

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

Good L-factors:
Prime L-Factor
$$2$$$$1 + 2 T + 3 T^{2} + 4 T^{3} + 4 T^{4}$$
$$5$$$$1 + 4 T + 12 T^{2} + 20 T^{3} + 25 T^{4}$$
$$11$$$$( 1 + 2 T + 11 T^{2} )^{2}$$
$$13$$$$1 + 8 T + 40 T^{2} + 104 T^{3} + 169 T^{4}$$
$$17$$$$1 + 4 T + 20 T^{2} + 68 T^{3} + 289 T^{4}$$
$$19$$$$1 + 30 T^{2} + 361 T^{4}$$
$$23$$$$1 + 4 T + 18 T^{2} + 92 T^{3} + 529 T^{4}$$
$$29$$$$1 + 8 T + 66 T^{2} + 232 T^{3} + 841 T^{4}$$
$\cdots$$\cdots$

## Sato-Tate group

$$\mathrm{ST} =$$ $G_{3,3}$, $$\quad \mathrm{ST}^0 = \mathrm{SU}(2)\times\mathrm{SU}(2)$$

## Endomorphisms of the Jacobian

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

Endomorphism algebra over $$\Q$$:

 $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{2})$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$

All $$\overline{\Q}$$-endomorphisms of the Jacobian are defined over $$\Q$$.

More complete information on endomorphism algebras and rings can be found on the pages of the individual curves in the isogeny class.