Properties

Label 3.2.ac_b_a
Base Field $\F_{2}$
Dimension $3$
Ordinary No
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{2}$
Dimension:  $3$
L-polynomial:  $1 - 2 x + x^{2} + 2 x^{4} - 8 x^{5} + 8 x^{6}$
Frobenius angles:  $\pm0.0693533547550$, $\pm0.339907131295$, $\pm0.770553776540$
Angle rank:  $2$ (numerical)
Number field:  6.0.2580992.1
Galois group:  $D_{6}$
Jacobians:  1

This isogeny class is simple and geometrically simple.

Newton polygon

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

Point counts

This isogeny class contains the Jacobians of 1 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 2 44 386 5984 16742 212300 2105602 16683392 164730518 1094290604

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 1 3 7 23 11 51 127 255 619 1043

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
The endomorphism algebra of this simple isogeny class is 6.0.2580992.1.
All geometric endomorphisms are defined over $\F_{2}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.2.c_b_a$2$3.4.ac_f_am