Properties

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

Learn more about

Invariants

Base field:  $\F_{2}$
Dimension:  $3$
L-polynomial:  $( 1 - x + 2 x^{2} )( 1 - 3 x + 5 x^{2} - 6 x^{3} + 4 x^{4} )$
Frobenius angles:  $\pm0.123548644961$, $\pm0.384973271919$, $\pm0.456881978294$
Angle rank:  $2$ (numerical)
Jacobians:  0

This isogeny class is not simple.

Newton polygon

This isogeny class is ordinary.

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

Point counts

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

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 2 152 1064 2736 21142 323456 3131242 20142432 135386552 1078157432

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 9 14 9 19 78 181 305 518 1029

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
The isogeny class factors as 1.2.ab $\times$ 2.2.ad_f and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{2}$
The base change of $A$ to $\F_{2^{6}}$ is 1.64.aj $\times$ 1.64.l 2 . The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{2^{6}}$.
Remainder of endomorphism lattice by field

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
3.2.ac_e_ah$2$3.4.e_e_ab
3.2.c_e_h$2$3.4.e_e_ab
3.2.e_k_r$2$3.4.e_e_ab
3.2.ab_b_b$3$3.8.f_t_cd
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.2.ac_e_ah$2$3.4.e_e_ab
3.2.c_e_h$2$3.4.e_e_ab
3.2.e_k_r$2$3.4.e_e_ab
3.2.ab_b_b$3$3.8.f_t_cd
3.2.b_b_ab$6$(not in LMFDB)
3.2.ab_d_ab$12$(not in LMFDB)
3.2.b_d_b$12$(not in LMFDB)