Properties

Label 4.3.ai_bf_adb_fx
Base Field $\F_{3}$
Dimension $4$
Ordinary No
$p$-rank $3$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{3}$
Dimension:  $4$
L-polynomial:  $( 1 - 3 x + 3 x^{2} )( 1 - 5 x + 13 x^{2} - 25 x^{3} + 39 x^{4} - 45 x^{5} + 27 x^{6} )$
Frobenius angles:  $\pm0.0714477711956$, $\pm0.166666666667$, $\pm0.272071776080$, $\pm0.560185743604$
Angle rank:  $3$ (numerical)

This isogeny class is not simple.

Newton polygon

$p$-rank:  $3$
Slopes:  $[0, 0, 0, 1/2, 1/2, 1, 1, 1]$

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
$A(\F_{q^r})$ 5 5425 457940 44078125 4326589525 300106399600 21852060841280 1845111414453125 152838760198077380 12202985346955860625

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -4 8 23 84 296 773 2089 6532 20039 59268

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad $\times$ 3.3.af_n_az 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 $\times$ 3.729.al_cmn_aoxb. 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
4.3.c_b_b_d$2$(not in LMFDB)
4.3.i_bf_db_fx$2$(not in LMFDB)
4.3.af_q_abo_da$3$(not in LMFDB)
4.3.ac_b_ab_d$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
4.3.c_b_b_d$2$(not in LMFDB)
4.3.i_bf_db_fx$2$(not in LMFDB)
4.3.af_q_abo_da$3$(not in LMFDB)
4.3.ac_b_ab_d$3$(not in LMFDB)
4.3.af_q_abo_da$6$(not in LMFDB)
4.3.f_q_bo_da$6$(not in LMFDB)