Properties

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

Learn more about

Invariants

Base field:  $\F_{421}$
Dimension:  $1$
L-polynomial:  $1 - 28 x + 421 x^{2}$
Frobenius angles:  $\pm0.260971855616$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-1}) \)
Galois group:  $C_2$
Jacobians:  13

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 13 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 394 177300 74631874 31414723200 13225454830954 5567914691363700 2344092095345709634 986862773183057164800 415469227534928891459914 174912544792462296261532500

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 394 177300 74631874 31414723200 13225454830954 5567914691363700 2344092095345709634 986862773183057164800 415469227534928891459914 174912544792462296261532500

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{421}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-1}) \).
All geometric endomorphisms are defined over $\F_{421}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
1.421.bc$2$(not in LMFDB)
1.421.abe$4$(not in LMFDB)
1.421.be$4$(not in LMFDB)