Properties

Label 2.3.af_m
Base Field $\F_{3}$
Dimension $2$
Ordinary No
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian No

Learn more about

Invariants

Base field:  $\F_{3}$
Dimension:  $2$
L-polynomial:  $( 1 - 3 x + 3 x^{2} )( 1 - 2 x + 3 x^{2} )$
Frobenius angles:  $\pm0.166666666667$, $\pm0.304086723985$
Angle rank:  $1$ (numerical)
Jacobians:  0

This isogeny class is not simple.

Newton polygon

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

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 84 1064 8736 65582 536256 4769438 43365504 391199816 3500898324

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 9 38 105 269 738 2183 6609 19874 59289

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad $\times$ 1.3.ac 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 $\overline{\F}_{3}$
The base change of $A$ to $\F_{3^{6}}$ is 1.729.abu $\times$ 1.729.cc. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{3^{6}}$.
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
2.3.ab_a$2$2.9.ab_m
2.3.b_a$2$2.9.ab_m
2.3.f_m$2$2.9.ab_m
2.3.ac_g$3$2.27.k_cc
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.3.ab_a$2$2.9.ab_m
2.3.b_a$2$2.9.ab_m
2.3.f_m$2$2.9.ab_m
2.3.ac_g$3$2.27.k_cc
2.3.ab_a$6$2.729.i_abnm
2.3.c_g$6$2.729.i_abnm