Properties

Label 2.4.ab_ab
Base field $\F_{2^{2}}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
Simple yes
Geometrically simple yes
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

Related objects

Downloads

Learn more

Invariants

Base field:  $\F_{2^{2}}$
Dimension:  $2$
L-polynomial:  $1 - x - x^{2} - 4 x^{3} + 16 x^{4}$
Frobenius angles:  $\pm0.153921966489$, $\pm0.719142225765$
Angle rank:  $2$ (numerical)
Number field:  4.0.45177.1
Galois group:  $D_{4}$
Jacobians:  $2$
Isomorphism classes:  2

This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $2$
Slopes:  $[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $11$ $231$ $3212$ $76923$ $1079441$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $4$ $14$ $49$ $298$ $1054$ $4151$ $16888$ $65554$ $262417$ $1050014$

Jacobians and polarizations

This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{2}}$.

Endomorphism algebra over $\F_{2^{2}}$
The endomorphism algebra of this simple isogeny class is 4.0.45177.1.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
2.4.b_ab$2$2.16.ad_z