# Properties

 Label 3.16.ar_fc_ayq Base Field $\F_{2^{4}}$ Dimension $3$ Ordinary No $p$-rank $1$ Principally polarizable Yes

## Invariants

 Base field: $\F_{2^{4}}$ Dimension: $3$ L-polynomial: $( 1 - 4 x )^{2}( 1 - 9 x + 44 x^{2} - 144 x^{3} + 256 x^{4} )$ Frobenius angles: $0$, $0$, $\pm0.126935807746$, $\pm0.434779740724$ Angle rank: $2$ (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 it is unknown whether it contains a Jacobian.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 1332 15118200 67046618268 277852348321200 1149673953169893492 4722521494376492188200 19345008997101125059065612 79228324808799966288846856800 324515925920136781841780221379172 1329225974335180086667206748722855000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 0 232 3996 64688 1045620 16777768 268465932 4294976096 68718920292 1099509955672

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
 The isogeny class factors as 1.16.ai $\times$ 2.16.aj_bs 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_bs : 4.0.40293.1.
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_am_cm $2$ (not in LMFDB) 3.16.b_am_acm $2$ (not in LMFDB) 3.16.r_fc_yq $2$ (not in LMFDB) 3.16.af_y_aei $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.16.ab_am_cm $2$ (not in LMFDB) 3.16.b_am_acm $2$ (not in LMFDB) 3.16.r_fc_yq $2$ (not in LMFDB) 3.16.af_y_aei $3$ (not in LMFDB) 3.16.aj_ci_alc $4$ (not in LMFDB) 3.16.j_ci_lc $4$ (not in LMFDB) 3.16.an_ds_arw $6$ (not in LMFDB) 3.16.f_y_ei $6$ (not in LMFDB) 3.16.n_ds_rw $6$ (not in LMFDB)