Properties

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

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Invariants

Base field:  $\F_{3}$
Dimension:  $3$
L-polynomial:  $( 1 - 3 x + 3 x^{2} )( 1 - x + 2 x^{2} - 3 x^{3} + 9 x^{4} )$
Frobenius angles:  $\pm0.166666666667$, $\pm0.235082516458$, $\pm0.648854628963$
Angle rank:  $2$ (numerical)
Jacobians:  2

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 contains the Jacobians of 2 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 8 896 17024 792064 19535848 396591104 10717358296 285171554304 7431198793088 204231505874816

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 0 10 24 114 320 748 2240 6626 19176 58570

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad $\times$ 2.3.ab_c 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.abk_bas. 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.ac_c_a$2$3.9.a_q_g
3.3.c_c_a$2$3.9.a_q_g
3.3.e_i_m$2$3.9.a_q_g
3.3.ab_f_ag$3$(not in LMFDB)
3.3.c_c_a$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.3.ac_c_a$2$3.9.a_q_g
3.3.c_c_a$2$3.9.a_q_g
3.3.e_i_m$2$3.9.a_q_g
3.3.ab_f_ag$3$(not in LMFDB)
3.3.c_c_a$3$(not in LMFDB)
3.3.ac_c_a$6$(not in LMFDB)
3.3.ab_f_ag$6$(not in LMFDB)
3.3.b_f_g$6$(not in LMFDB)