Properties

Label 3.16.ar_ey_axk
Base Field $\F_{2^{4}}$
Dimension $3$
Ordinary No
$p$-rank $1$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{2^{4}}$
Dimension:  $3$
L-polynomial:  $( 1 - 4 x )^{4}( 1 - x + 16 x^{2} )$
Frobenius angles:  $0$, $0$, $0$, $0$, $\pm0.460106912325$
Angle rank:  $1$ (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 does not contain a Jacobian.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1296 14580000 65280270384 275208376680000 1147110230217915216 4719439187419729500000 19339906587766422714212976 79222025989104838191798480000 324511364919066730624322566145424 1329223716733347248285389835362500000

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 0 224 3888 64064 1043280 16766816 268395120 4294634624 68717954448 1099508088224

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
The isogeny class factors as 1.16.ai 2 $\times$ 1.16.ab 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 isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{4}}$.

SubfieldPrimitive Model
$\F_{2}$3.2.ab_ac_e
$\F_{2}$3.2.ab_g_ae
$\F_{2}$3.2.b_ac_ae
$\F_{2}$3.2.b_g_e

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
3.16.ap_ds_aqa$2$(not in LMFDB)
3.16.ab_aq_bg$2$(not in LMFDB)
3.16.b_aq_abg$2$(not in LMFDB)
3.16.p_ds_qa$2$(not in LMFDB)
3.16.r_ey_xk$2$(not in LMFDB)
3.16.af_u_aey$3$(not in LMFDB)
3.16.h_ce_ia$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.16.ap_ds_aqa$2$(not in LMFDB)
3.16.ab_aq_bg$2$(not in LMFDB)
3.16.b_aq_abg$2$(not in LMFDB)
3.16.p_ds_qa$2$(not in LMFDB)
3.16.r_ey_xk$2$(not in LMFDB)
3.16.af_u_aey$3$(not in LMFDB)
3.16.h_ce_ia$3$(not in LMFDB)
3.16.aj_ce_alc$4$(not in LMFDB)
3.16.ah_bo_aiq$4$(not in LMFDB)
3.16.ab_bw_abg$4$(not in LMFDB)
3.16.b_bw_bg$4$(not in LMFDB)
3.16.h_bo_iq$4$(not in LMFDB)
3.16.j_ce_lc$4$(not in LMFDB)
3.16.d_bc_ei$5$(not in LMFDB)
3.16.an_do_arg$6$(not in LMFDB)
3.16.al_cq_ami$6$(not in LMFDB)
3.16.aj_cu_als$6$(not in LMFDB)
3.16.ah_ce_aia$6$(not in LMFDB)
3.16.ad_m_aey$6$(not in LMFDB)
3.16.ab_bg_aq$6$(not in LMFDB)
3.16.b_bg_q$6$(not in LMFDB)
3.16.d_m_ey$6$(not in LMFDB)
3.16.f_u_ey$6$(not in LMFDB)
3.16.j_cu_ls$6$(not in LMFDB)
3.16.l_cq_mi$6$(not in LMFDB)
3.16.n_do_rg$6$(not in LMFDB)
3.16.ab_q_a$8$(not in LMFDB)
3.16.b_q_a$8$(not in LMFDB)
3.16.af_bk_afo$10$(not in LMFDB)
3.16.ad_bc_aei$10$(not in LMFDB)
3.16.f_bk_fo$10$(not in LMFDB)
3.16.af_ca_age$12$(not in LMFDB)
3.16.ad_bs_ads$12$(not in LMFDB)
3.16.ab_a_q$12$(not in LMFDB)
3.16.b_a_aq$12$(not in LMFDB)
3.16.d_bs_ds$12$(not in LMFDB)
3.16.f_ca_ge$12$(not in LMFDB)