# Properties

 Label 3.16.at_gf_abfc 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: $( 1 - 4 x )^{2}( 1 - 11 x + 57 x^{2} - 176 x^{3} + 256 x^{4} )$ Frobenius angles: $0$, $0$, $\pm0.0728689886706$, $\pm0.368631800070$ 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 $A(\F_{q^r})$ 1143 14316075 66931501392 278133063001875 1147765479839878383 4717796890056573960000 19341117087390655008627303 79228754581661894026424851875 324518583758652220931538185228112 1329225728567999629128944922737476875

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -2 218 3991 64754 1043878 16760975 268411918 4294999394 68719483111 1099509752378

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
 The isogeny class factors as 1.16.ai $\times$ 2.16.al_cf 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_cf : 4.0.78057.3.
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_ap_ea $2$ (not in LMFDB) 3.16.d_ap_aea $2$ (not in LMFDB) 3.16.t_gf_bfc $2$ (not in LMFDB) 3.16.ah_bd_aeu $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.16.ad_ap_ea $2$ (not in LMFDB) 3.16.d_ap_aea $2$ (not in LMFDB) 3.16.t_gf_bfc $2$ (not in LMFDB) 3.16.ah_bd_aeu $3$ (not in LMFDB) 3.16.al_cv_ano $4$ (not in LMFDB) 3.16.l_cv_no $4$ (not in LMFDB) 3.16.ap_en_awi $6$ (not in LMFDB) 3.16.h_bd_eu $6$ (not in LMFDB) 3.16.p_en_wi $6$ (not in LMFDB)