Properties

Label 3.16.ar_fc_ayq
Base Field $\F_{2^{4}}$
Dimension $3$
Ordinary No
$p$-rank $1$
Principally polarizable Yes

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Invariants

Base field:  $\F_{2^{4}}$
Dimension:  $3$
L-polynomial:  $( 1 - 4 x )^{2}( 1 - 9 x + 44 x^{2} - 144 x^{3} + 256 x^{4} )$
Frobenius angles:  $0$, $0$, $\pm0.126935807746$, $\pm0.434779740724$
Angle rank:  $2$ (numerical)

This isogeny class is not simple.

Newton polygon

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

Point counts

This isogeny class is principally polarizable, but it is unknown whether it contains a Jacobian.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1332 15118200 67046618268 277852348321200 1149673953169893492 4722521494376492188200 19345008997101125059065612 79228324808799966288846856800 324515925920136781841780221379172 1329225974335180086667206748722855000

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 0 232 3996 64688 1045620 16777768 268465932 4294976096 68718920292 1099509955672

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
The isogeny class factors as 1.16.ai $\times$ 2.16.aj_bs and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{2^{4}}$.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
3.16.ab_am_cm$2$(not in LMFDB)
3.16.b_am_acm$2$(not in LMFDB)
3.16.r_fc_yq$2$(not in LMFDB)
3.16.af_y_aei$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.16.ab_am_cm$2$(not in LMFDB)
3.16.b_am_acm$2$(not in LMFDB)
3.16.r_fc_yq$2$(not in LMFDB)
3.16.af_y_aei$3$(not in LMFDB)
3.16.aj_ci_alc$4$(not in LMFDB)
3.16.j_ci_lc$4$(not in LMFDB)
3.16.an_ds_arw$6$(not in LMFDB)
3.16.f_y_ei$6$(not in LMFDB)
3.16.n_ds_rw$6$(not in LMFDB)