Properties

Label 3.7.aj_bq_afb
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:  $( 1 - 5 x + 7 x^{2} )( 1 - 4 x + 15 x^{2} - 28 x^{3} + 49 x^{4} )$
Frobenius angles:  $\pm0.106147807505$, $\pm0.250816204349$, $\pm0.483874642948$
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
$A(\F_{q^r})$ 99 124839 41955408 13714937379 4788093539889 1641348592595712 558939213215845407 191417022372347246475 65709450332466262269552 22544387121482673630410799

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 53 356 2381 16949 118580 824123 5759861 40351820 282538493

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7}$
The isogeny class factors as 1.7.af $\times$ 2.7.ae_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.ab_c_at$2$(not in LMFDB)
3.7.b_c_t$2$(not in LMFDB)
3.7.j_bq_fb$2$(not in LMFDB)
3.7.ad_s_abp$3$(not in LMFDB)
3.7.a_g_e$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.7.ab_c_at$2$(not in LMFDB)
3.7.b_c_t$2$(not in LMFDB)
3.7.j_bq_fb$2$(not in LMFDB)
3.7.ad_s_abp$3$(not in LMFDB)
3.7.a_g_e$3$(not in LMFDB)
3.7.ai_bm_aem$6$(not in LMFDB)
3.7.af_ba_act$6$(not in LMFDB)
3.7.a_g_ae$6$(not in LMFDB)
3.7.d_s_bp$6$(not in LMFDB)
3.7.f_ba_ct$6$(not in LMFDB)
3.7.i_bm_em$6$(not in LMFDB)