Properties

Label 3.16.ar_fc_ayq
Base field $\F_{2^{4}}$
Dimension $3$
$p$-rank $1$
Ordinary no
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes

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Invariants

Base field:  $\F_{2^{4}}$
Dimension:  $3$
L-polynomial:  $( 1 - 4 x )^{2}( 1 - 9 x + 44 x^{2} - 144 x^{3} + 256 x^{4} )$
  $1 - 17 x + 132 x^{2} - 640 x^{3} + 2112 x^{4} - 4352 x^{5} + 4096 x^{6}$
Frobenius angles:  $0$, $0$, $\pm0.126935807746$, $\pm0.434779740724$
Angle rank:  $2$ (numerical)

This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $1332$ $15118200$ $67046618268$ $277852348321200$ $1149673953169893492$

Point counts of the (virtual) curve

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

Jacobians and polarizations

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{4}}$.

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:

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)