Properties

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

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Invariants

Base field:  $\F_{3}$
Dimension:  $3$
L-polynomial:  $( 1 - 2 x + 3 x^{2} )( 1 - 2 x + x^{2} - 6 x^{3} + 9 x^{4} )$
Frobenius angles:  $\pm0.0292466093486$, $\pm0.304086723985$, $\pm0.637420057318$
Angle rank:  $1$ (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
$A(\F_{q^r})$ 6 684 12312 530784 14114166 320013504 9669223122 282430166400 7552438699464 205891158689964

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 0 10 18 82 240 592 2016 6562 19494 59050

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ac $\times$ 2.3.ac_b 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 3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-2}) \)$)$
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.a_a_ak$2$3.9.a_a_abu
3.3.a_a_k$2$3.9.a_a_abu
3.3.e_i_o$2$3.9.a_a_abu
3.3.c_f_e$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.3.a_a_ak$2$3.9.a_a_abu
3.3.a_a_k$2$3.9.a_a_abu
3.3.e_i_o$2$3.9.a_a_abu
3.3.c_f_e$3$(not in LMFDB)
3.3.ag_v_abs$6$(not in LMFDB)
3.3.ac_f_ae$6$(not in LMFDB)
3.3.g_v_bs$6$(not in LMFDB)
3.3.ac_b_e$12$(not in LMFDB)
3.3.c_b_ae$12$(not in LMFDB)
3.3.ag_t_abo$24$(not in LMFDB)
3.3.ac_d_ai$24$(not in LMFDB)
3.3.c_d_i$24$(not in LMFDB)
3.3.g_t_bo$24$(not in LMFDB)