# Properties

 Label 1.421.a Base Field $\F_{421}$ Dimension $1$ Ordinary No $p$-rank $0$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{421}$ Dimension: $1$ L-polynomial: $1 + 421 x^{2}$ Frobenius angles: $\pm0.5$ Angle rank: $0$ (numerical) Number field: $$\Q(\sqrt{-421})$$ Galois group: $C_2$ Jacobians: 10

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 10 curves, and hence is principally polarizable:

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 422 178084 74618462 31414017600 13225450646102 5567914871245444 2344092097965587342 986862773180683526400 415469227535518665906182 174912544792479809247794404

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 422 178084 74618462 31414017600 13225450646102 5567914871245444 2344092097965587342 986862773180683526400 415469227535518665906182 174912544792479809247794404

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{421}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-421})$$.
Endomorphism algebra over $\overline{\F}_{421}$
 The base change of $A$ to $\F_{421^{2}}$ is the simple isogeny class 1.177241.bgk and its endomorphism algebra is the quaternion algebra over $$\Q$$ ramified at $421$ and $\infty$.
All geometric endomorphisms are defined over $\F_{421^{2}}$.

## Base change

This is a primitive isogeny class.

## Twists

This isogeny class has no twists.