Properties

Label 3.7.ak_bz_agk
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:  $1 - 10 x + 51 x^{2} - 166 x^{3} + 357 x^{4} - 490 x^{5} + 343 x^{6}$
Frobenius angles:  $\pm0.0683199847700$, $\pm0.284796870547$, $\pm0.407332955184$
Angle rank:  $3$ (numerical)
Number field:  6.0.12178624.1
Galois group:  $S_4\times C_2$

This isogeny class is simple and geometrically 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
$A(\F_{q^r})$ 86 121948 44162978 13762075696 4658150804726 1618368845763772 558329821626191666 191611914119074964224 65713443926415447470702 22540570795576448403639388

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -2 52 376 2388 16488 116920 823226 5765724 40354270 282490672

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7}$
The endomorphism algebra of this simple isogeny class is 6.0.12178624.1.
All geometric endomorphisms are defined over $\F_{7}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.7.k_bz_gk$2$(not in LMFDB)