# Properties

 Label 3.7.ak_bz_agi 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: $( 1 - 4 x + 7 x^{2} )( 1 - 6 x + 20 x^{2} - 42 x^{3} + 49 x^{4} )$ Frobenius angles: $\pm0.147692939668$, $\pm0.227185525829$, $\pm0.422977188212$ 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 $A(\F_{q^r})$ 88 124608 45269224 14229236736 4795443628888 1642465590397632 560650205860272616 191638464260035313664 65685461865479142622936 22535759737562919570833088

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -2 52 382 2468 16978 118660 826642 5766524 40337086 282430372

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7}$
 The isogeny class factors as 1.7.ae $\times$ 2.7.ag_u 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.ac_d_ae $2$ (not in LMFDB) 3.7.c_d_e $2$ (not in LMFDB) 3.7.k_bz_gi $2$ (not in LMFDB) 3.7.ah_bh_aea $3$ (not in LMFDB) 3.7.ab_ad_q $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.7.ac_d_ae $2$ (not in LMFDB) 3.7.c_d_e $2$ (not in LMFDB) 3.7.k_bz_gi $2$ (not in LMFDB) 3.7.ah_bh_aea $3$ (not in LMFDB) 3.7.ab_ad_q $3$ (not in LMFDB) 3.7.al_cf_ahc $6$ (not in LMFDB) 3.7.af_v_acm $6$ (not in LMFDB) 3.7.b_ad_aq $6$ (not in LMFDB) 3.7.f_v_cm $6$ (not in LMFDB) 3.7.h_bh_ea $6$ (not in LMFDB) 3.7.l_cf_hc $6$ (not in LMFDB)