Properties

Label 3.16.ar_fj_abau
Base Field $\F_{2^{4}}$
Dimension $3$
Ordinary No
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{2^{4}}$
Dimension:  $3$
L-polynomial:  $( 1 - 4 x )^{2}( 1 - 9 x + 51 x^{2} - 144 x^{3} + 256 x^{4} )$
Frobenius angles:  $0$, $0$, $\pm0.252176979752$, $\pm0.361066333219$
Angle rank:  $2$ (numerical)

This isogeny class is not simple.

Newton polygon

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

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
$A(\F_{q^r})$ 1395 16077375 70178984820 281852927368875 1150364380978828125 4717970060505776802000 19338908194663776137457495 79225601524842002622828289875 324516191006253163271665512772620 1329224769975505927004896663037109375

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 0 246 4185 65626 1046250 16761591 268381260 4294828466 68718976425 1099508959446

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
The isogeny class factors as 1.16.ai $\times$ 2.16.aj_bz 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_af_eq$2$(not in LMFDB)
3.16.b_af_aeq$2$(not in LMFDB)
3.16.r_fj_bau$2$(not in LMFDB)
3.16.af_bf_adg$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.16.ab_af_eq$2$(not in LMFDB)
3.16.b_af_aeq$2$(not in LMFDB)
3.16.r_fj_bau$2$(not in LMFDB)
3.16.af_bf_adg$3$(not in LMFDB)
3.16.aj_cp_alc$4$(not in LMFDB)
3.16.j_cp_lc$4$(not in LMFDB)
3.16.an_dz_asy$6$(not in LMFDB)
3.16.f_bf_dg$6$(not in LMFDB)
3.16.n_dz_sy$6$(not in LMFDB)