# 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

## 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.

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

 $r$ 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: 1.16.ai 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 1.16.ab : $$\Q(\sqrt{-7})$$.
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 $\F_{2^{4}}$.

 Subfield Primitive 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.
 Twist Extension Degree Common 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.
 Twist Extension Degree Common 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)