Properties

Label 3.2.ac_c_ab
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 yes

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Invariants

Base field:  $\F_{2}$
Dimension:  $3$
L-polynomial:  $1 - 2 x + 2 x^{2} - x^{3} + 4 x^{4} - 8 x^{5} + 8 x^{6}$
Frobenius angles:  $\pm0.161334789180$, $\pm0.327009058845$, $\pm0.739882802642$
Angle rank:  $3$ (numerical)
Number field:  6.0.2464727.1
Galois group:  $S_4\times C_2$
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:  $3$
Slopes:  $[0, 0, 0, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $4$ $104$ $592$ $11024$ $31604$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $1$ $5$ $10$ $33$ $31$ $62$ $155$ $257$ $586$ $1065$

Jacobians and polarizations

This isogeny class contains the Jacobians of 2 curves (of which 1 is hyperelliptic), and hence is principally polarizable:

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 6.0.2464727.1.

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_c_b$2$3.4.a_i_ab