Properties

Label 3.7.aj_bo_aer
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:  $( 1 - 5 x + 7 x^{2} )( 1 - 4 x + 13 x^{2} - 28 x^{3} + 49 x^{4} )$
Frobenius angles:  $\pm0.106147807505$, $\pm0.204545263622$, $\pm0.514205363720$
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
$A(\F_{q^r})$ 93 114855 39211776 13626971475 4845127076823 1648826948321280 559282060852288041 191507412525777976875 65722777207740542585088 22541737097470480089861375

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 49 332 2365 17149 119116 824627 5762581 40360004 282505289

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7}$
The isogeny class factors as 1.7.af $\times$ 2.7.ae_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.ab_a_aj$2$(not in LMFDB)
3.7.b_a_j$2$(not in LMFDB)
3.7.j_bo_er$2$(not in LMFDB)
3.7.ad_q_abr$3$(not in LMFDB)
3.7.a_e_ae$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.7.ab_a_aj$2$(not in LMFDB)
3.7.b_a_j$2$(not in LMFDB)
3.7.j_bo_er$2$(not in LMFDB)
3.7.ad_q_abr$3$(not in LMFDB)
3.7.a_e_ae$3$(not in LMFDB)
3.7.ai_bk_aee$6$(not in LMFDB)
3.7.af_y_acr$6$(not in LMFDB)
3.7.a_e_e$6$(not in LMFDB)
3.7.d_q_br$6$(not in LMFDB)
3.7.f_y_cr$6$(not in LMFDB)
3.7.i_bk_ee$6$(not in LMFDB)