# Properties

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

## Invariants

 Base field: $\F_{2^{4}}$ Dimension: $3$ L-polynomial: $( 1 - 5 x + 16 x^{2} )( 1 - 7 x + 16 x^{2} )^{2}$ Frobenius angles: $\pm0.160861246510$, $\pm0.160861246510$, $\pm0.285098958592$ Angle rank: $2$ (numerical)

This isogeny class is not simple.

## Newton polygon

This isogeny class is ordinary.

 $p$-rank: $3$ Slopes: $[0, 0, 0, 1, 1, 1]$

## 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 $A(\F_{q^r})$ 1200 15206400 70458757200 285408921600000 1157130647865750000 4725534502537151846400 19344809110947560958061200 79229627008386207993446400000 324519871691574786700836054385200 1329228561530838549088135276704000000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -2 230 4198 66446 1052398 16788470 268463158 4295046686 68719755838 1099512095750

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
 The isogeny class factors as 1.16.ah 2 $\times$ 1.16.af 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.ah 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-15})$$$)$ 1.16.af : $$\Q(\sqrt{-39})$$.
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.aj_bb_abr $2$ (not in LMFDB) 3.16.af_ab_dh $2$ (not in LMFDB) 3.16.f_ab_adh $2$ (not in LMFDB) 3.16.j_bb_br $2$ (not in LMFDB) 3.16.t_gl_bgv $2$ (not in LMFDB) 3.16.c_o_ch $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.16.aj_bb_abr $2$ (not in LMFDB) 3.16.af_ab_dh $2$ (not in LMFDB) 3.16.f_ab_adh $2$ (not in LMFDB) 3.16.j_bb_br $2$ (not in LMFDB) 3.16.t_gl_bgv $2$ (not in LMFDB) 3.16.c_o_ch $3$ (not in LMFDB) 3.16.af_bh_adh $4$ (not in LMFDB) 3.16.f_bh_dh $4$ (not in LMFDB) 3.16.am_dg_aoz $6$ (not in LMFDB) 3.16.ac_o_ach $6$ (not in LMFDB) 3.16.m_dg_oz $6$ (not in LMFDB)