Properties

Label 3.7.ak_bx_afz
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:  $( 1 - 5 x + 7 x^{2} )( 1 - 5 x + 17 x^{2} - 35 x^{3} + 49 x^{4} )$
Frobenius angles:  $\pm0.106147807505$, $\pm0.197751856397$, $\pm0.457936209148$
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
$A(\F_{q^r})$ 81 112671 41176836 13670259759 4783070555856 1647056804558256 560822888561694147 191604190204656248175 65697713371940048165268 22539958845783200991726336

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -2 48 349 2372 16933 118989 826894 5765492 40344613 282483003

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7}$
The isogeny class factors as 1.7.af $\times$ 2.7.af_r 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_ab_ap$2$(not in LMFDB)
3.7.a_ab_p$2$(not in LMFDB)
3.7.k_bx_fz$2$(not in LMFDB)
3.7.ae_t_acb$3$(not in LMFDB)
3.7.ab_e_ac$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.7.a_ab_ap$2$(not in LMFDB)
3.7.a_ab_p$2$(not in LMFDB)
3.7.k_bx_fz$2$(not in LMFDB)
3.7.ae_t_acb$3$(not in LMFDB)
3.7.ab_e_ac$3$(not in LMFDB)
3.7.aj_bs_afi$6$(not in LMFDB)
3.7.ag_bd_adj$6$(not in LMFDB)
3.7.b_e_c$6$(not in LMFDB)
3.7.e_t_cb$6$(not in LMFDB)
3.7.g_bd_dj$6$(not in LMFDB)
3.7.j_bs_fi$6$(not in LMFDB)