Properties

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

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Invariants

Base field:  $\F_{2^{4}}$
Dimension:  $3$
L-polynomial:  $( 1 - 4 x )^{2}( 1 - 9 x + 49 x^{2} - 144 x^{3} + 256 x^{4} )$
Frobenius angles:  $0$, $0$, $\pm0.211195784157$, $\pm0.390536017683$
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 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
$A(\F_{q^r})$ 1377 15801075 69278307588 280790174895075 1150658848649701017 4720510706003044243200 19342194515954253793416537 79226624212052108294365092675 324513606021701173373959610258052 1329220656346904506770262566981901875

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 0 242 4131 65378 1046520 16770623 268426872 4294883906 68718429027 1099505556722

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
The isogeny class factors as 1.16.ai $\times$ 2.16.aj_bx 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_ah_ea$2$(not in LMFDB)
3.16.b_ah_aea$2$(not in LMFDB)
3.16.r_fh_bae$2$(not in LMFDB)
3.16.af_bd_ado$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.16.ab_ah_ea$2$(not in LMFDB)
3.16.b_ah_aea$2$(not in LMFDB)
3.16.r_fh_bae$2$(not in LMFDB)
3.16.af_bd_ado$3$(not in LMFDB)
3.16.aj_cn_alc$4$(not in LMFDB)
3.16.j_cn_lc$4$(not in LMFDB)
3.16.an_dx_asq$6$(not in LMFDB)
3.16.f_bd_do$6$(not in LMFDB)
3.16.n_dx_sq$6$(not in LMFDB)