Properties

Label 3.7.aj_bn_aem
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:  $( 1 - 5 x + 7 x^{2} )( 1 - 4 x + 12 x^{2} - 28 x^{3} + 49 x^{4} )$
Frobenius angles:  $\pm0.106147807505$, $\pm0.182041207691$, $\pm0.527071640754$
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
$A(\F_{q^r})$ 90 109980 37878840 13551735600 4856642851950 1649887421757120 559509476998517730 191621020997352960000 65738473828071997222440 22541475295756331716779900

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 47 320 2351 17189 119192 824963 5765999 40369640 282502007

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7}$
The isogeny class factors as 1.7.af $\times$ 2.7.ae_m 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_ab_ae$2$(not in LMFDB)
3.7.b_ab_e$2$(not in LMFDB)
3.7.j_bn_em$2$(not in LMFDB)
3.7.ad_p_abs$3$(not in LMFDB)
3.7.a_d_ai$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.7.ab_ab_ae$2$(not in LMFDB)
3.7.b_ab_e$2$(not in LMFDB)
3.7.j_bn_em$2$(not in LMFDB)
3.7.ad_p_abs$3$(not in LMFDB)
3.7.a_d_ai$3$(not in LMFDB)
3.7.ai_bj_aea$6$(not in LMFDB)
3.7.af_x_acq$6$(not in LMFDB)
3.7.a_d_i$6$(not in LMFDB)
3.7.d_p_bs$6$(not in LMFDB)
3.7.f_x_cq$6$(not in LMFDB)
3.7.i_bj_ea$6$(not in LMFDB)