Properties

Label 3.7.aj_bn_aem
Base field $\F_{7}$
Dimension $3$
$p$-rank $3$
Ordinary yes
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian no

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Invariants

Base field:  $\F_{7}$
Dimension:  $3$
L-polynomial:  $( 1 - 5 x + 7 x^{2} )( 1 - 4 x + 12 x^{2} - 28 x^{3} + 49 x^{4} )$
  $1 - 9 x + 39 x^{2} - 116 x^{3} + 273 x^{4} - 441 x^{5} + 343 x^{6}$
Frobenius angles:  $\pm0.106147807505$, $\pm0.182041207691$, $\pm0.527071640754$
Angle rank:  $3$ (numerical)
Isomorphism classes:  12

This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $3$
Slopes:  $[0, 0, 0, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $90$ $109980$ $37878840$ $13551735600$ $4856642851950$

Point counts of the (virtual) curve

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

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{7}$.

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:

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)