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

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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.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 88 124608 45269224 14229236736 4795443628888 1642465590397632 560650205860272616 191638464260035313664 65685461865479142622936 22535759737562919570833088

Point counts of the (virtual) curve

$r$ 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.
TwistExtension DegreeCommon 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.
TwistExtension DegreeCommon 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)