Properties

Label 3.16.ar_fg_azw
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:  $( 1 - 4 x )^{2}( 1 - 9 x + 48 x^{2} - 144 x^{3} + 256 x^{4} )$
Frobenius angles:  $0$, $0$, $\pm0.193865395619$, $\pm0.401408407366$
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
$A(\F_{q^r})$ 1368 15663600 68829731208 280234867442400 1150658754712211448 4721415846238358139600 19343429234499307042087272 79227264065306987278064894400 324513497775436678815426172922328 1329220439318938514568431325801390000

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 0 240 4104 65248 1046520 16773840 268444008 4294918592 68718406104 1099505377200

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
The isogeny class factors as 1.16.ai $\times$ 2.16.aj_bw 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_ai_ds$2$(not in LMFDB)
3.16.b_ai_ads$2$(not in LMFDB)
3.16.r_fg_zw$2$(not in LMFDB)
3.16.af_bc_ads$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.16.ab_ai_ds$2$(not in LMFDB)
3.16.b_ai_ads$2$(not in LMFDB)
3.16.r_fg_zw$2$(not in LMFDB)
3.16.af_bc_ads$3$(not in LMFDB)
3.16.aj_cm_alc$4$(not in LMFDB)
3.16.j_cm_lc$4$(not in LMFDB)
3.16.an_dw_asm$6$(not in LMFDB)
3.16.f_bc_ds$6$(not in LMFDB)
3.16.n_dw_sm$6$(not in LMFDB)