Properties

Label 3.3.ae_j_aq
Base Field $\F_{3}$
Dimension $3$
Ordinary Yes
$p$-rank $3$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{3}$
Dimension:  $3$
L-polynomial:  $( 1 - 2 x + 3 x^{2} )( 1 - 2 x + 2 x^{2} - 6 x^{3} + 9 x^{4} )$
Frobenius angles:  $\pm0.116139763599$, $\pm0.304086723985$, $\pm0.616139763599$
Angle rank:  $2$ (numerical)
Jacobians:  4

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 contains the Jacobians of 4 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 8 960 16568 614400 16708648 363833280 10174091992 294794035200 7793263245896 207571534104000

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 0 12 24 92 280 684 2128 6844 20112 59532

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ac $\times$ 2.3.ac_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 $\F_{3^{4}}$ is 1.81.ac 2 $\times$ 1.81.o. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{3^{4}}$.
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.a_b_ai$2$3.9.c_h_ae
3.3.a_b_i$2$3.9.c_h_ae
3.3.e_j_q$2$3.9.c_h_ae
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.3.a_b_ai$2$3.9.c_h_ae
3.3.a_b_i$2$3.9.c_h_ae
3.3.e_j_q$2$3.9.c_h_ae
3.3.ac_ab_i$8$(not in LMFDB)
3.3.ac_h_ai$8$(not in LMFDB)
3.3.c_ab_ai$8$(not in LMFDB)
3.3.c_h_i$8$(not in LMFDB)