# Properties

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

## Invariants

 Base field: $\F_{2^{4}}$ Dimension: $3$ L-polynomial: $( 1 - 4 x )^{2}( 1 - 9 x + 49 x^{2} - 144 x^{3} + 256 x^{4} )$ Frobenius angles: $0$, $0$, $\pm0.211195784157$, $\pm0.390536017683$ 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 it is unknown whether it contains a Jacobian.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1377 15801075 69278307588 280790174895075 1150658848649701017 4720510706003044243200 19342194515954253793416537 79226624212052108294365092675 324513606021701173373959610258052 1329220656346904506770262566981901875

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 0 242 4131 65378 1046520 16770623 268426872 4294883906 68718429027 1099505556722

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
 The isogeny class factors as 1.16.ai $\times$ 2.16.aj_bx 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_bx : 4.0.2873.1.
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_ah_ea $2$ (not in LMFDB) 3.16.b_ah_aea $2$ (not in LMFDB) 3.16.r_fh_bae $2$ (not in LMFDB) 3.16.af_bd_ado $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.16.ab_ah_ea $2$ (not in LMFDB) 3.16.b_ah_aea $2$ (not in LMFDB) 3.16.r_fh_bae $2$ (not in LMFDB) 3.16.af_bd_ado $3$ (not in LMFDB) 3.16.aj_cn_alc $4$ (not in LMFDB) 3.16.j_cn_lc $4$ (not in LMFDB) 3.16.an_dx_asq $6$ (not in LMFDB) 3.16.f_bd_do $6$ (not in LMFDB) 3.16.n_dx_sq $6$ (not in LMFDB)