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

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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.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 87 123279 44715564 13994755359 4726539147312 1630429805165616 559536102533813289 191651465905654779375 65708754933069306113292 22540474672271024195660544

Point counts of the (virtual) curve

$r$ 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.
TwistExtension DegreeCommon 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.
TwistExtension DegreeCommon 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)