Properties

Label 3.3.af_n_ay
Base Field $\F_{3}$
Dimension $3$
Ordinary No
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{3}$
Dimension:  $3$
L-polynomial:  $( 1 - 3 x + 3 x^{2} )( 1 - 2 x + 4 x^{2} - 6 x^{3} + 9 x^{4} )$
Frobenius angles:  $\pm0.166666666667$, $\pm0.210767374595$, $\pm0.567777800232$
Angle rank:  $2$ (numerical)
Jacobians:  0

This isogeny class is not simple.

Newton polygon

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

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
$A(\F_{q^r})$ 6 924 19656 624624 20196546 447999552 10434001038 283069602816 7624602557208 202574870579004

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 11 26 95 329 836 2183 6575 19682 58091

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad $\times$ 2.3.ac_e 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.cc $\times$ 2.729.ca_czy. 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
3.3.ab_b_a$2$3.9.b_h_bq
3.3.f_n_y$2$3.9.b_h_bq
3.3.ac_h_am$3$(not in LMFDB)
3.3.b_b_a$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.3.ab_b_a$2$3.9.b_h_bq
3.3.f_n_y$2$3.9.b_h_bq
3.3.ac_h_am$3$(not in LMFDB)
3.3.b_b_a$3$(not in LMFDB)
3.3.ac_h_am$6$(not in LMFDB)
3.3.c_h_m$6$(not in LMFDB)