# Properties

 Label 363.a Conductor $363$ Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q \times \Q$$ $$\End(J) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

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## Genus 2 curves in isogeny class 363.a

Label Equation
363.a.11979.1 $$y^2 + (x^2 + 1)y = x^5 + 2x^3 + 4x^2 + 2x$$
363.a.43923.1 $$y^2 + x^2y = 11x^5 - 13x^4 - 7x^3 + 10x^2 + x - 2$$

## L-function data

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

Bad L-factors:
Prime L-Factor
$$3$$$$( 1 + T )( 1 + T + 3 T^{2} )$$
$$11$$$$( 1 - T )^{2}$$

Good L-factors:
Prime L-Factor
$$2$$$$( 1 - T + 2 T^{2} )( 1 + 2 T + 2 T^{2} )$$
$$5$$$$( 1 - T + 5 T^{2} )( 1 + 2 T + 5 T^{2} )$$
$$7$$$$( 1 - 4 T + 7 T^{2} )( 1 + 2 T + 7 T^{2} )$$
$$13$$$$( 1 - 4 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )$$
$$17$$$$( 1 + 2 T + 17 T^{2} )^{2}$$
$$19$$$$( 1 + 19 T^{2} )^{2}$$
$$23$$$$( 1 - 8 T + 23 T^{2} )( 1 + T + 23 T^{2} )$$
$$29$$$$( 1 + 29 T^{2} )( 1 + 6 T + 29 T^{2} )$$
$\cdots$$\cdots$

See L-function page for more information

## Sato-Tate group

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

## Decomposition of the Jacobian

Splits over $$\Q$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
Elliptic curve isogeny class 33.a
Elliptic curve isogeny class 11.a

## Endomorphisms of the Jacobian

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

Endomorphism algebra over $$\Q$$:

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

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.