# Properties

 Label 1.337.a Base Field $\F_{337}$ Dimension $1$ Ordinary No $p$-rank $0$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{337}$ Dimension: $1$ L-polynomial: $1 + 337 x^{2}$ Frobenius angles: $\pm0.5$ Angle rank: $0$ (numerical) Number field: $$\Q(\sqrt{-337})$$ Galois group: $C_2$ Jacobians: 8

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 8 curves, and hence is principally polarizable:

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 338 114244 38272754 12897690624 4346598285458 1464803698744516 493638820681066034 166356282543723417600 56062067225927988301778 18892916655146425254269764

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 338 114244 38272754 12897690624 4346598285458 1464803698744516 493638820681066034 166356282543723417600 56062067225927988301778 18892916655146425254269764

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{337}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-337})$$.
Endomorphism algebra over $\overline{\F}_{337}$
 The base change of $A$ to $\F_{337^{2}}$ is the simple isogeny class 1.113569.zy and its endomorphism algebra is the quaternion algebra over $$\Q$$ ramified at $337$ and $\infty$.
All geometric endomorphisms are defined over $\F_{337^{2}}$.

## Base change

This is a primitive isogeny class.

## Twists

This isogeny class has no twists.