# Properties

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

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## Invariants

 Base field: $\F_{2^{4}}$ Dimension: $3$ L-polynomial: $( 1 - 4 x )^{2}( 1 - 11 x + 61 x^{2} - 176 x^{3} + 256 x^{4} )$ Frobenius angles: $0$, $0$, $\pm0.189901625224$, $\pm0.315486115946$ 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 $A(\F_{q^r})$ 1179 14884875 69139408464 282369785083875 1152349828029027639 4719972182006527992000 19339785506519610466147299 79225780152744446567367859875 324516733024657024925157473527824 1329225292830976931805349917919921875

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -2 226 4123 65746 1048058 16768711 268393438 4294838146 68719091203 1099509391946

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
 The isogeny class factors as 1.16.ai $\times$ 2.16.al_cj 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_cj : 4.0.22625.1.
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_al_fg $2$ (not in LMFDB) 3.16.d_al_afg $2$ (not in LMFDB) 3.16.t_gj_bgi $2$ (not in LMFDB) 3.16.ah_bh_aee $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.16.ad_al_fg $2$ (not in LMFDB) 3.16.d_al_afg $2$ (not in LMFDB) 3.16.t_gj_bgi $2$ (not in LMFDB) 3.16.ah_bh_aee $3$ (not in LMFDB) 3.16.al_cz_ano $4$ (not in LMFDB) 3.16.l_cz_no $4$ (not in LMFDB) 3.16.ap_er_awy $6$ (not in LMFDB) 3.16.h_bh_ee $6$ (not in LMFDB) 3.16.p_er_wy $6$ (not in LMFDB)