Properties

Label 3.16.ar_ez_axs
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:  $( 1 - 4 x )^{2}( 1 - 9 x + 41 x^{2} - 144 x^{3} + 256 x^{4} )$
Frobenius angles:  $0$, $0$, $\pm0.0608845495576$, $\pm0.454248801212$
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
$A(\F_{q^r})$ 1305 14713875 65720368980 275894116999875 1147899425817065625 4720614293017310688000 19341903946139014747337505 79224658626489099271265383875 324514068088727343111863446440180 1329226648038512961508119216076171875

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 0 226 3915 64226 1044000 16770991 268422840 4294777346 68718526875 1099510512946

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
The isogeny class factors as 1.16.ai $\times$ 2.16.aj_bp 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_ap_bo$2$(not in LMFDB)
3.16.b_ap_abo$2$(not in LMFDB)
3.16.r_ez_xs$2$(not in LMFDB)
3.16.af_v_aeu$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.16.ab_ap_bo$2$(not in LMFDB)
3.16.b_ap_abo$2$(not in LMFDB)
3.16.r_ez_xs$2$(not in LMFDB)
3.16.af_v_aeu$3$(not in LMFDB)
3.16.aj_cf_alc$4$(not in LMFDB)
3.16.j_cf_lc$4$(not in LMFDB)
3.16.an_dp_ark$6$(not in LMFDB)
3.16.f_v_eu$6$(not in LMFDB)
3.16.n_dp_rk$6$(not in LMFDB)