# Properties

 Label 3.16.at_ge_abeu 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: $( 1 - 4 x )^{4}( 1 - 3 x + 16 x^{2} )$ Frobenius angles: $0$, $0$, $0$, $0$, $\pm0.377642706461$ 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 $A(\F_{q^r})$ 1134 14175000 66382977654 277034980950000 1146319479249574014 4716211009130036325000 19339119766848606045878214 79225739313395830480716300000 324514375995661382014727289471054 1329220989977712956601982041796875000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -2 216 3958 64496 1042558 16755336 268384198 4294835936 68718592078 1099505832696

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
 The isogeny class factors as 1.16.ai 2 $\times$ 1.16.ad 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.ad : $$\Q(\sqrt{-55})$$.
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.an_cm_aiq $2$ (not in LMFDB) 3.16.ad_aq_ds $2$ (not in LMFDB) 3.16.d_aq_ads $2$ (not in LMFDB) 3.16.n_cm_iq $2$ (not in LMFDB) 3.16.t_ge_beu $2$ (not in LMFDB) 3.16.ah_bc_aey $3$ (not in LMFDB) 3.16.f_bo_ei $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.16.an_cm_aiq $2$ (not in LMFDB) 3.16.ad_aq_ds $2$ (not in LMFDB) 3.16.d_aq_ads $2$ (not in LMFDB) 3.16.n_cm_iq $2$ (not in LMFDB) 3.16.t_ge_beu $2$ (not in LMFDB) 3.16.ah_bc_aey $3$ (not in LMFDB) 3.16.f_bo_ei $3$ (not in LMFDB) 3.16.al_cu_ano $4$ (not in LMFDB) 3.16.af_y_age $4$ (not in LMFDB) 3.16.ad_bw_ads $4$ (not in LMFDB) 3.16.d_bw_ds $4$ (not in LMFDB) 3.16.f_y_ge $4$ (not in LMFDB) 3.16.l_cu_no $4$ (not in LMFDB) 3.16.b_u_dc $5$ (not in LMFDB) 3.16.ap_em_awe $6$ (not in LMFDB) 3.16.al_dk_apk $6$ (not in LMFDB) 3.16.aj_bs_ahk $6$ (not in LMFDB) 3.16.af_bo_aei $6$ (not in LMFDB) 3.16.ad_bg_abw $6$ (not in LMFDB) 3.16.ab_e_aey $6$ (not in LMFDB) 3.16.b_e_ey $6$ (not in LMFDB) 3.16.d_bg_bw $6$ (not in LMFDB) 3.16.h_bc_ey $6$ (not in LMFDB) 3.16.j_bs_hk $6$ (not in LMFDB) 3.16.l_dk_pk $6$ (not in LMFDB) 3.16.p_em_we $6$ (not in LMFDB) 3.16.ad_q_a $8$ (not in LMFDB) 3.16.d_q_a $8$ (not in LMFDB) 3.16.ah_bs_agu $10$ (not in LMFDB) 3.16.ab_u_adc $10$ (not in LMFDB) 3.16.h_bs_gu $10$ (not in LMFDB) 3.16.ah_ci_aiq $12$ (not in LMFDB) 3.16.ad_a_bw $12$ (not in LMFDB) 3.16.ab_bk_abg $12$ (not in LMFDB) 3.16.b_bk_bg $12$ (not in LMFDB) 3.16.d_a_abw $12$ (not in LMFDB) 3.16.h_ci_iq $12$ (not in LMFDB)