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

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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.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 75 102375 37770300 13270359375 4783635606000 1650012929838000 560965499956055325 191720589960793359375 65740026518341194570300 22545316578869898559200000

Point counts of the (virtual) curve

$r$ 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.
TwistExtension DegreeCommon 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.
TwistExtension DegreeCommon 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)