Properties

Label 3.2.ac_d_af
Base field $\F_{2}$
Dimension $3$
$p$-rank $3$
Ordinary yes
Supersingular no
Simple yes
Geometrically simple yes
Primitive yes
Principally polarizable yes
Contains a Jacobian no

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Invariants

Base field:  $\F_{2}$
Dimension:  $3$
L-polynomial:  $1 - 2 x + 3 x^{2} - 5 x^{3} + 6 x^{4} - 8 x^{5} + 8 x^{6}$
Frobenius angles:  $\pm0.0964297006873$, $\pm0.413303350042$, $\pm0.672715745192$
Angle rank:  $3$ (numerical)
Number field:  6.0.3194271.1
Galois group:  $S_4\times C_2$
Jacobians:  $0$
Isomorphism classes:  1

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

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $3$
Slopes:  $[0, 0, 0, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $3$ $99$ $243$ $4059$ $26763$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $1$ $7$ $4$ $15$ $26$ $52$ $176$ $311$ $481$ $1072$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{2}$
The endomorphism algebra of this simple isogeny class is 6.0.3194271.1.

Base change

This is a primitive isogeny class.

Twists

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

TwistExtension degreeCommon base change
3.2.c_d_f$2$3.4.c_b_af