# Properties

 Label 1.227.a Base Field $\F_{227}$ Dimension $1$ Ordinary No $p$-rank $0$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{227}$ Dimension: $1$ L-polynomial: $1 + 227 x^{2}$ Frobenius angles: $\pm0.5$ Angle rank: $0$ (numerical) Number field: $$\Q(\sqrt{-227})$$ Galois group: $C_2$ Jacobians: 20

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 20 curves, and hence is principally polarizable:

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 228 51984 11697084 2655134784 602738989908 136821774103056 31058537410917804 7050287986967865600 1600415374247183470788 363294289955316125848464

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 228 51984 11697084 2655134784 602738989908 136821774103056 31058537410917804 7050287986967865600 1600415374247183470788 363294289955316125848464

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{227}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-227})$$.
Endomorphism algebra over $\overline{\F}_{227}$
 The base change of $A$ to $\F_{227^{2}}$ is the simple isogeny class 1.51529.rm and its endomorphism algebra is the quaternion algebra over $$\Q$$ ramified at $227$ and $\infty$.
All geometric endomorphisms are defined over $\F_{227^{2}}$.

## Base change

This is a primitive isogeny class.

## Twists

This isogeny class has no twists.