Properties

Label 3.7.ak_bt_aff
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:  $( 1 - 5 x + 7 x^{2} )( 1 - 5 x + 13 x^{2} - 35 x^{3} + 49 x^{4} )$
Frobenius angles:  $\pm0.0616448849068$, $\pm0.106147807505$, $\pm0.511587336964$
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
$A(\F_{q^r})$ 69 92391 34480404 12787745919 4727062720944 1637900701675056 558825243763515063 191566927803691316175 65743058244179560877892 22545439165532314764884736

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -2 40 289 2212 16733 118333 823954 5764372 40372453 282551675

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7}$
The isogeny class factors as 1.7.af $\times$ 2.7.af_n 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_af_af$2$(not in LMFDB)
3.7.a_af_f$2$(not in LMFDB)
3.7.k_bt_ff$2$(not in LMFDB)
3.7.ae_p_acf$3$(not in LMFDB)
3.7.ab_a_as$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.7.a_af_af$2$(not in LMFDB)
3.7.a_af_f$2$(not in LMFDB)
3.7.k_bt_ff$2$(not in LMFDB)
3.7.ae_p_acf$3$(not in LMFDB)
3.7.ab_a_as$3$(not in LMFDB)
3.7.aj_bo_aes$6$(not in LMFDB)
3.7.ag_z_adf$6$(not in LMFDB)
3.7.b_a_s$6$(not in LMFDB)
3.7.e_p_cf$6$(not in LMFDB)
3.7.g_z_df$6$(not in LMFDB)
3.7.j_bo_es$6$(not in LMFDB)