Properties

Label 3.16.ar_ff_azn
Base Field $\F_{2^{4}}$
Dimension $3$
Ordinary Yes
$p$-rank $3$
Principally polarizable Yes

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Invariants

Base field:  $\F_{2^{4}}$
Dimension:  $3$
L-polynomial:  $( 1 - 7 x + 16 x^{2} )( 1 - 10 x + 49 x^{2} - 160 x^{3} + 256 x^{4} )$
Frobenius angles:  $\pm0.0660425289118$, $\pm0.160861246510$, $\pm0.412497962872$
Angle rank:  $3$ (numerical)

This isogeny class is not simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $3$
Slopes:  $[0, 0, 0, 1, 1, 1]$

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
$A(\F_{q^r})$ 1360 15536640 68444775760 279959936471040 1151407619028994000 4723739663040646940160 19346928220938637171199440 79231299381048171516249538560 324518148975527722013913000354640 1329226029373747921257879483034176000

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 0 238 4080 65182 1047200 16782094 268492560 4295137342 68719391040 1099510001198

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
The isogeny class factors as 1.16.ah $\times$ 2.16.ak_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 is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.16.ad_af_x$2$(not in LMFDB)
3.16.d_af_ax$2$(not in LMFDB)
3.16.r_ff_zn$2$(not in LMFDB)