# Properties

 Label 1.401.a Base Field $\F_{401}$ Dimension $1$ Ordinary No $p$-rank $0$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{401}$ Dimension: $1$ L-polynomial: $1 + 401 x^{2}$ Frobenius angles: $\pm0.5$ Angle rank: $0$ (numerical) Number field: $$\Q(\sqrt{-401})$$ 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 $A(\F_{q^r})$ 402 161604 64481202 25856640000 10368641602002 4157825411364804 1667287938243362802 668582463183874560000 268101567757470981763602 107508728670766600970408004

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 402 161604 64481202 25856640000 10368641602002 4157825411364804 1667287938243362802 668582463183874560000 268101567757470981763602 107508728670766600970408004

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{401}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-401})$$.
Endomorphism algebra over $\overline{\F}_{401}$
 The base change of $A$ to $\F_{401^{2}}$ is the simple isogeny class 1.160801.bew and its endomorphism algebra is the quaternion algebra over $$\Q$$ ramified at $401$ and $\infty$.
All geometric endomorphisms are defined over $\F_{401^{2}}$.

## Base change

This is a primitive isogeny class.

## Twists

This isogeny class has no twists.