# Properties

 Label 3.7.ak_bx_afz 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: $( 1 - 5 x + 7 x^{2} )( 1 - 5 x + 17 x^{2} - 35 x^{3} + 49 x^{4} )$ Frobenius angles: $\pm0.106147807505$, $\pm0.197751856397$, $\pm0.457936209148$ 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 $A(\F_{q^r})$ 81 112671 41176836 13670259759 4783070555856 1647056804558256 560822888561694147 191604190204656248175 65697713371940048165268 22539958845783200991726336

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -2 48 349 2372 16933 118989 826894 5765492 40344613 282483003

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7}$
 The isogeny class factors as 1.7.af $\times$ 2.7.af_r 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_ab_ap $2$ (not in LMFDB) 3.7.a_ab_p $2$ (not in LMFDB) 3.7.k_bx_fz $2$ (not in LMFDB) 3.7.ae_t_acb $3$ (not in LMFDB) 3.7.ab_e_ac $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.7.a_ab_ap $2$ (not in LMFDB) 3.7.a_ab_p $2$ (not in LMFDB) 3.7.k_bx_fz $2$ (not in LMFDB) 3.7.ae_t_acb $3$ (not in LMFDB) 3.7.ab_e_ac $3$ (not in LMFDB) 3.7.aj_bs_afi $6$ (not in LMFDB) 3.7.ag_bd_adj $6$ (not in LMFDB) 3.7.b_e_c $6$ (not in LMFDB) 3.7.e_t_cb $6$ (not in LMFDB) 3.7.g_bd_dj $6$ (not in LMFDB) 3.7.j_bs_fi $6$ (not in LMFDB)