# Properties

 Label 1.19.a Base Field $\F_{19}$ Dimension $1$ Ordinary No $p$-rank $0$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{19}$ Dimension: $1$ L-polynomial: $1 + 19 x^{2}$ Frobenius angles: $\pm0.5$ Angle rank: $0$ (numerical) Number field: $$\Q(\sqrt{-19})$$ Galois group: $C_2$ Jacobians: 4

This isogeny class is simple and geometrically simple.

## Newton polygon

This isogeny class is supersingular. $p$-rank: $0$ Slopes: $[1/2, 1/2]$

## Point counts

This isogeny class contains the Jacobians of 4 curves, and hence is principally polarizable:

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 20 400 6860 129600 2476100 47059600 893871740 16983302400 322687697780 6131071210000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 20 400 6860 129600 2476100 47059600 893871740 16983302400 322687697780 6131071210000

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{19}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-19})$$.
Endomorphism algebra over $\overline{\F}_{19}$
 The base change of $A$ to $\F_{19^{2}}$ is the simple isogeny class 1.361.bm and its endomorphism algebra is the quaternion algebra over $$\Q$$ ramified at $19$ and $\infty$.
All geometric endomorphisms are defined over $\F_{19^{2}}$.

## Base change

This is a primitive isogeny class.

## Twists

This isogeny class has no twists.