# Properties

 Label 3.16.ar_fj_abau 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: $( 1 - 4 x )^{2}( 1 - 9 x + 51 x^{2} - 144 x^{3} + 256 x^{4} )$ Frobenius angles: $0$, $0$, $\pm0.252176979752$, $\pm0.361066333219$ Angle rank: $2$ (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 $A(\F_{q^r})$ 1395 16077375 70178984820 281852927368875 1150364380978828125 4717970060505776802000 19338908194663776137457495 79225601524842002622828289875 324516191006253163271665512772620 1329224769975505927004896663037109375

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 0 246 4185 65626 1046250 16761591 268381260 4294828466 68718976425 1099508959446

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
 The isogeny class factors as 1.16.ai $\times$ 2.16.aj_bz 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 $\infty$. 2.16.aj_bz : 4.0.42625.1.
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.ab_af_eq $2$ (not in LMFDB) 3.16.b_af_aeq $2$ (not in LMFDB) 3.16.r_fj_bau $2$ (not in LMFDB) 3.16.af_bf_adg $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.16.ab_af_eq $2$ (not in LMFDB) 3.16.b_af_aeq $2$ (not in LMFDB) 3.16.r_fj_bau $2$ (not in LMFDB) 3.16.af_bf_adg $3$ (not in LMFDB) 3.16.aj_cp_alc $4$ (not in LMFDB) 3.16.j_cp_lc $4$ (not in LMFDB) 3.16.an_dz_asy $6$ (not in LMFDB) 3.16.f_bf_dg $6$ (not in LMFDB) 3.16.n_dz_sy $6$ (not in LMFDB)