# Properties

 Label 3.7.ak_bv_afp 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: $( 1 - 5 x + 7 x^{2} )( 1 - 5 x + 15 x^{2} - 35 x^{3} + 49 x^{4} )$ Frobenius angles: $\pm0.106147807505$, $\pm0.139519842760$, $\pm0.487441680688$ 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 $A(\F_{q^r})$ 75 102375 37770300 13270359375 4783635606000 1650012929838000 560965499956055325 191720589960793359375 65740026518341194570300 22545316578869898559200000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -2 44 319 2300 16933 119201 827104 5768996 40370593 282550139

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7}$
 The isogeny class factors as 1.7.af $\times$ 2.7.af_p 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_ad_af $2$ (not in LMFDB) 3.7.a_ad_f $2$ (not in LMFDB) 3.7.k_bv_fp $2$ (not in LMFDB) 3.7.ae_r_acd $3$ (not in LMFDB) 3.7.ab_c_ak $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.7.a_ad_af $2$ (not in LMFDB) 3.7.a_ad_f $2$ (not in LMFDB) 3.7.k_bv_fp $2$ (not in LMFDB) 3.7.ae_r_acd $3$ (not in LMFDB) 3.7.ab_c_ak $3$ (not in LMFDB) 3.7.aj_bq_afa $6$ (not in LMFDB) 3.7.ag_bb_adh $6$ (not in LMFDB) 3.7.b_c_k $6$ (not in LMFDB) 3.7.e_r_cd $6$ (not in LMFDB) 3.7.g_bb_dh $6$ (not in LMFDB) 3.7.j_bq_fa $6$ (not in LMFDB)