# Properties

 Label 3.7.ak_bw_afu Base Field $\F_{7}$ Dimension $3$ Ordinary Yes $p$-rank $3$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{7}$ Dimension: $3$ L-polynomial: $( 1 - 5 x + 7 x^{2} )( 1 - 5 x + 16 x^{2} - 35 x^{3} + 49 x^{4} )$ Frobenius angles: $\pm0.106147807505$, $\pm0.169178782589$, $\pm0.473594973839$ Angle rank: $3$ (numerical)

This isogeny class is not simple.

## Newton polygon

This isogeny class is ordinary.

 $p$-rank: $3$ Slopes: $[0, 0, 0, 1, 1, 1]$

## 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 $A(\F_{q^r})$ 78 107484 39458016 13480213344 4790434219038 1650333728848896 561108653042933346 191668447132197360000 65717329386644766893088 22542263711395652400454044

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -2 46 334 2338 16958 119224 827314 5767426 40356658 282511886

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7}$
 The isogeny class factors as 1.7.af $\times$ 2.7.af_q and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{7}$.

## 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 3.7.a_ac_ak $2$ (not in LMFDB) 3.7.a_ac_k $2$ (not in LMFDB) 3.7.k_bw_fu $2$ (not in LMFDB) 3.7.ae_s_acc $3$ (not in LMFDB) 3.7.ab_d_ag $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.7.a_ac_ak $2$ (not in LMFDB) 3.7.a_ac_k $2$ (not in LMFDB) 3.7.k_bw_fu $2$ (not in LMFDB) 3.7.ae_s_acc $3$ (not in LMFDB) 3.7.ab_d_ag $3$ (not in LMFDB) 3.7.aj_br_afe $6$ (not in LMFDB) 3.7.ag_bc_adi $6$ (not in LMFDB) 3.7.b_d_g $6$ (not in LMFDB) 3.7.e_s_cc $6$ (not in LMFDB) 3.7.g_bc_di $6$ (not in LMFDB) 3.7.j_br_fe $6$ (not in LMFDB)