# Properties

 Label 3.16.ar_fg_azw 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: $( 1 - 4 x )^{2}( 1 - 9 x + 48 x^{2} - 144 x^{3} + 256 x^{4} )$ Frobenius angles: $0$, $0$, $\pm0.193865395619$, $\pm0.401408407366$ 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 $A(\F_{q^r})$ 1368 15663600 68829731208 280234867442400 1150658754712211448 4721415846238358139600 19343429234499307042087272 79227264065306987278064894400 324513497775436678815426172922328 1329220439318938514568431325801390000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 0 240 4104 65248 1046520 16773840 268444008 4294918592 68718406104 1099505377200

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
 The isogeny class factors as 1.16.ai $\times$ 2.16.aj_bw 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_bw : 4.0.21964.1.
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_ai_ds $2$ (not in LMFDB) 3.16.b_ai_ads $2$ (not in LMFDB) 3.16.r_fg_zw $2$ (not in LMFDB) 3.16.af_bc_ads $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.16.ab_ai_ds $2$ (not in LMFDB) 3.16.b_ai_ads $2$ (not in LMFDB) 3.16.r_fg_zw $2$ (not in LMFDB) 3.16.af_bc_ads $3$ (not in LMFDB) 3.16.aj_cm_alc $4$ (not in LMFDB) 3.16.j_cm_lc $4$ (not in LMFDB) 3.16.an_dw_asm $6$ (not in LMFDB) 3.16.f_bc_ds $6$ (not in LMFDB) 3.16.n_dw_sm $6$ (not in LMFDB)