# Properties

 Label 3.16.aw_ib_abqe 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: $( 1 - 4 x )^{2}( 1 - 7 x + 16 x^{2} )^{2}$ Frobenius angles: $0$, $0$, $\pm0.160861246510$, $\pm0.160861246510$ 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 $A(\F_{q^r})$ 900 12960000 66393828900 281192653440000 1154347715633062500 4724644459624820640000 19344808894726080836532900 79228745681805484124981760000 324516886945453578183169879836900 1329223763698450236495207202500000000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -5 191 3955 65471 1049875 16785311 268463155 4294998911 68719123795 1099508127071

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
 The isogeny class factors as 1.16.ai $\times$ 1.16.ah 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.ah 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-15})$$$)$
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_ab_fg $2$ (not in LMFDB) 3.16.ag_ap_hs $2$ (not in LMFDB) 3.16.g_ap_ahs $2$ (not in LMFDB) 3.16.i_ab_afg $2$ (not in LMFDB) 3.16.w_ib_bqe $2$ (not in LMFDB) 3.16.ak_bp_aeu $3$ (not in LMFDB) 3.16.ab_ah_abo $3$ (not in LMFDB) 3.16.l_cz_ns $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.16.ai_ab_fg $2$ (not in LMFDB) 3.16.ag_ap_hs $2$ (not in LMFDB) 3.16.g_ap_ahs $2$ (not in LMFDB) 3.16.i_ab_afg $2$ (not in LMFDB) 3.16.w_ib_bqe $2$ (not in LMFDB) 3.16.ak_bp_aeu $3$ (not in LMFDB) 3.16.ab_ah_abo $3$ (not in LMFDB) 3.16.l_cz_ns $3$ (not in LMFDB) 3.16.ao_dt_arg $4$ (not in LMFDB) 3.16.ai_bh_afg $4$ (not in LMFDB) 3.16.a_ab_a $4$ (not in LMFDB) 3.16.a_bh_a $4$ (not in LMFDB) 3.16.i_bh_fg $4$ (not in LMFDB) 3.16.o_dt_rg $4$ (not in LMFDB) 3.16.as_fx_abds $6$ (not in LMFDB) 3.16.ap_eb_asu $6$ (not in LMFDB) 3.16.al_cz_ans $6$ (not in LMFDB) 3.16.ae_ab_cq $6$ (not in LMFDB) 3.16.ad_v_ado $6$ (not in LMFDB) 3.16.b_ah_bo $6$ (not in LMFDB) 3.16.d_v_do $6$ (not in LMFDB) 3.16.e_ab_acq $6$ (not in LMFDB) 3.16.k_bp_eu $6$ (not in LMFDB) 3.16.p_eb_su $6$ (not in LMFDB) 3.16.s_fx_bds $6$ (not in LMFDB) 3.16.ah_bx_aiq $12$ (not in LMFDB) 3.16.ae_bh_acq $12$ (not in LMFDB) 3.16.e_bh_cq $12$ (not in LMFDB) 3.16.h_bx_iq $12$ (not in LMFDB)