# Properties

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

## Invariants

 Base field: $\F_{2^{4}}$ Dimension: $3$ L-polynomial: $( 1 - 4 x )^{2}( 1 - 5 x + 16 x^{2} )^{2}$ Frobenius angles: $0$, $0$, $\pm0.285098958592$, $\pm0.285098958592$ Angle rank: $1$ (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.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1296 15681600 70413806736 283248900000000 1151715733086598416 4717229964795458510400 19335731077125739182817296 79222671724663902243600000000 324517095502749275604122886158736 1329229985557187271158741753787240000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -1 239 4199 65951 1047479 16758959 268337159 4294669631 68719167959 1099513273679

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
 The isogeny class factors as 1.16.ai $\times$ 1.16.af 2 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 : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$. 1.16.af 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-39})$$$)$
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.
 Twist Extension Degree Common base change 3.16.ai_x_ace $2$ (not in LMFDB) 3.16.ac_ah_fg $2$ (not in LMFDB) 3.16.c_ah_afg $2$ (not in LMFDB) 3.16.i_x_ce $2$ (not in LMFDB) 3.16.s_fx_bdw $2$ (not in LMFDB) 3.16.ag_bh_ado $3$ (not in LMFDB) 3.16.ad_ap_dk $3$ (not in LMFDB) 3.16.j_bt_ho $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.16.ai_x_ace $2$ (not in LMFDB) 3.16.ac_ah_fg $2$ (not in LMFDB) 3.16.c_ah_afg $2$ (not in LMFDB) 3.16.i_x_ce $2$ (not in LMFDB) 3.16.s_fx_bdw $2$ (not in LMFDB) 3.16.ag_bh_ado $3$ (not in LMFDB) 3.16.ad_ap_dk $3$ (not in LMFDB) 3.16.j_bt_ho $3$ (not in LMFDB) 3.16.ak_cv_ami $4$ (not in LMFDB) 3.16.ai_j_ce $4$ (not in LMFDB) 3.16.a_j_a $4$ (not in LMFDB) 3.16.a_x_a $4$ (not in LMFDB) 3.16.i_j_ace $4$ (not in LMFDB) 3.16.k_cv_mi $4$ (not in LMFDB) 3.16.ao_ej_avc $6$ (not in LMFDB) 3.16.an_cn_aiy $6$ (not in LMFDB) 3.16.aj_bt_aho $6$ (not in LMFDB) 3.16.ae_x_abc $6$ (not in LMFDB) 3.16.ab_f_aeu $6$ (not in LMFDB) 3.16.b_f_eu $6$ (not in LMFDB) 3.16.d_ap_adk $6$ (not in LMFDB) 3.16.e_x_bc $6$ (not in LMFDB) 3.16.g_bh_do $6$ (not in LMFDB) 3.16.n_cn_iy $6$ (not in LMFDB) 3.16.o_ej_vc $6$ (not in LMFDB) 3.16.af_z_age $12$ (not in LMFDB) 3.16.ae_j_bc $12$ (not in LMFDB) 3.16.e_j_abc $12$ (not in LMFDB) 3.16.f_z_ge $12$ (not in LMFDB)