# Properties

 Label 1.421.at Base Field $\F_{421}$ Dimension $1$ Ordinary Yes $p$-rank $1$ Principally polarizable Yes Contains a Jacobian Yes

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## Invariants

 Base field: $\F_{421}$ Dimension: $1$ L-polynomial: $1 - 19 x + 421 x^{2}$ Frobenius angles: $\pm0.346772388685$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-3})$$ Galois group: $C_2$ Jacobians: 10

This isogeny class is simple and geometrically simple.

## Newton polygon

This isogeny class is ordinary.

 $p$-rank: $1$ Slopes: $[0, 1]$

## 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 $A(\F_{q^r})$ 403 177723 74635600 31414495203 13225445770303 5567914577534400 2344092097273290403 986862773291182234083 415469227536715852224400 174912544792456035831906003

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 403 177723 74635600 31414495203 13225445770303 5567914577534400 2344092097273290403 986862773291182234083 415469227536715852224400 174912544792456035831906003

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{421}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-3})$$.
All geometric endomorphisms are defined over $\F_{421}$.

## Base change

This is a primitive isogeny class.

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 1.421.t $2$ (not in LMFDB) 1.421.aw $3$ (not in LMFDB) 1.421.bp $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 1.421.t $2$ (not in LMFDB) 1.421.aw $3$ (not in LMFDB) 1.421.bp $3$ (not in LMFDB) 1.421.abp $6$ (not in LMFDB) 1.421.w $6$ (not in LMFDB)