# Properties

 Label 3.16.at_gh_abfs 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: $( 1 - 4 x )^{2}( 1 - 11 x + 59 x^{2} - 176 x^{3} + 256 x^{4} )$ Frobenius angles: $0$, $0$, $\pm0.133878927982$, $\pm0.347077071791$ 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 $A(\F_{q^r})$ 1161 14599575 68032644876 280282363458075 1150297296203146311 4719711176171189125200 19342248765590222946866349 79229949044965879176813834675 324520349399954424887225543759124 1329226922797812775716769917551964375

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -2 222 4057 65258 1046188 16767783 268427626 4295064146 68719857001 1099510740222

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
 The isogeny class factors as 1.16.ai $\times$ 2.16.al_ch 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.al_ch : 4.0.90753.1.
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.ad_an_eq $2$ (not in LMFDB) 3.16.d_an_aeq $2$ (not in LMFDB) 3.16.t_gh_bfs $2$ (not in LMFDB) 3.16.ah_bf_aem $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.16.ad_an_eq $2$ (not in LMFDB) 3.16.d_an_aeq $2$ (not in LMFDB) 3.16.t_gh_bfs $2$ (not in LMFDB) 3.16.ah_bf_aem $3$ (not in LMFDB) 3.16.al_cx_ano $4$ (not in LMFDB) 3.16.l_cx_no $4$ (not in LMFDB) 3.16.ap_ep_awq $6$ (not in LMFDB) 3.16.h_bf_em $6$ (not in LMFDB) 3.16.p_ep_wq $6$ (not in LMFDB)