# Properties

 Label 3.7.ak_bz_agj 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: $( 1 - 5 x + 7 x^{2} )( 1 - 5 x + 19 x^{2} - 35 x^{3} + 49 x^{4} )$ Frobenius angles: $\pm0.106147807505$, $\pm0.260350433790$, $\pm0.415892662795$ 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 $A(\F_{q^r})$ 87 123279 44715564 13994755359 4726539147312 1630429805165616 559536102533813289 191651465905654779375 65708754933069306113292 22540474672271024195660544

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -2 52 379 2428 16733 117793 825004 5766916 40351393 282489467

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7}$
 The isogeny class factors as 1.7.af $\times$ 2.7.af_t 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_b_az $2$ (not in LMFDB) 3.7.a_b_z $2$ (not in LMFDB) 3.7.k_bz_gj $2$ (not in LMFDB) 3.7.ae_v_abz $3$ (not in LMFDB) 3.7.ab_g_g $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.7.a_b_az $2$ (not in LMFDB) 3.7.a_b_z $2$ (not in LMFDB) 3.7.k_bz_gj $2$ (not in LMFDB) 3.7.ae_v_abz $3$ (not in LMFDB) 3.7.ab_g_g $3$ (not in LMFDB) 3.7.aj_bu_afq $6$ (not in LMFDB) 3.7.ag_bf_adl $6$ (not in LMFDB) 3.7.b_g_ag $6$ (not in LMFDB) 3.7.e_v_bz $6$ (not in LMFDB) 3.7.g_bf_dl $6$ (not in LMFDB) 3.7.j_bu_fq $6$ (not in LMFDB)