Properties

Label 1.421.aw
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:  $1 - 22 x + 421 x^{2}$
Frobenius angles:  $\pm0.319894277981$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}) \)
Galois group:  $C_2$
Jacobians:  24

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 24 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 400 177600 74635600 31414598400 13225448410000 5567914577534400 2344092095728555600 986862773255121177600 415469227536715852224400 174912544792474809095640000

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 400 177600 74635600 31414598400 13225448410000 5567914577534400 2344092095728555600 986862773255121177600 415469227536715852224400 174912544792474809095640000

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.
TwistExtension DegreeCommon base change
1.421.w$2$(not in LMFDB)
1.421.at$3$(not in LMFDB)
1.421.bp$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
1.421.w$2$(not in LMFDB)
1.421.at$3$(not in LMFDB)
1.421.bp$3$(not in LMFDB)
1.421.abp$6$(not in LMFDB)
1.421.t$6$(not in LMFDB)