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

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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.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 78 107484 39458016 13480213344 4790434219038 1650333728848896 561108653042933346 191668447132197360000 65717329386644766893088 22542263711395652400454044

Point counts of the (virtual) curve

$r$ 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.
TwistExtension DegreeCommon 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.
TwistExtension DegreeCommon 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)