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:

Point counts of the abelian variety

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

Point counts of the curve

$r$ 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.
TwistExtension DegreeCommon 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.
TwistExtension DegreeCommon 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)