# Properties

 Label 1.277.a Base Field $\F_{277}$ Dimension $1$ Ordinary No $p$-rank $0$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{277}$ Dimension: $1$ L-polynomial: $1 + 277 x^{2}$ Frobenius angles: $\pm0.5$ Angle rank: $0$ (numerical) Number field: $$\Q(\sqrt{-277})$$ Galois group: $C_2$ Jacobians: 6

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 6 curves, and hence is principally polarizable:

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 278 77284 21253934 5887185984 1630793025158 451729710476356 125129118027271454 34660765681779513600 9601032097114511317238 2659485890903981220924964

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 278 77284 21253934 5887185984 1630793025158 451729710476356 125129118027271454 34660765681779513600 9601032097114511317238 2659485890903981220924964

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{277}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-277})$$.
Endomorphism algebra over $\overline{\F}_{277}$
 The base change of $A$ to $\F_{277^{2}}$ is the simple isogeny class 1.76729.vi and its endomorphism algebra is the quaternion algebra over $$\Q$$ ramified at $277$ and $\infty$.
All geometric endomorphisms are defined over $\F_{277^{2}}$.

## Base change

This is a primitive isogeny class.

## Twists

This isogeny class has no twists.