# Properties

 Label 3.16.as_fr_abca 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: $( 1 - 4 x )^{2}( 1 - 10 x + 51 x^{2} - 160 x^{3} + 256 x^{4} )$ Frobenius angles: $0$, $0$, $\pm0.118775077357$, $\pm0.396715540983$ 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 $A(\F_{q^r})$ 1242 14841900 67410885150 278530575717600 1149080543839229562 4720642471404865687500 19344294699311758106125422 79230199569895617471844502400 324517952317539894662554871999850 1329225045583212868416758049548947500

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -1 227 4019 64847 1045079 16771091 268456019 4295077727 68719349399 1099509187427

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{4}}$
 The isogeny class factors as 1.16.ai $\times$ 2.16.ak_bz 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.ak_bz : 4.0.281664.1.
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.ac_an_dk $2$ (not in LMFDB) 3.16.c_an_adk $2$ (not in LMFDB) 3.16.s_fr_bca $2$ (not in LMFDB) 3.16.ag_bb_aem $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.16.ac_an_dk $2$ (not in LMFDB) 3.16.c_an_adk $2$ (not in LMFDB) 3.16.s_fr_bca $2$ (not in LMFDB) 3.16.ag_bb_aem $3$ (not in LMFDB) 3.16.ak_cp_ami $4$ (not in LMFDB) 3.16.k_cp_mi $4$ (not in LMFDB) 3.16.ao_ed_aue $6$ (not in LMFDB) 3.16.g_bb_em $6$ (not in LMFDB) 3.16.o_ed_ue $6$ (not in LMFDB)