Properties

Label 3.7.aj_bm_aei
Base field $\F_{7}$
Dimension $3$
$p$-rank $2$
Ordinary no
Supersingular no
Simple yes
Geometrically simple yes
Primitive yes
Principally polarizable yes
Contains a Jacobian no

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Invariants

Base field:  $\F_{7}$
Dimension:  $3$
L-polynomial:  $1 - 9 x + 38 x^{2} - 112 x^{3} + 266 x^{4} - 441 x^{5} + 343 x^{6}$
Frobenius angles:  $\pm0.0283467889665$, $\pm0.193732293337$, $\pm0.536888824784$
Angle rank:  $3$ (numerical)
Number field:  6.0.30088184.1
Galois group:  $S_4\times C_2$
Isomorphism classes:  2

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

Newton polygon

$p$-rank:  $2$
Slopes:  $[0, 0, 1/2, 1/2, 1, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $86$ $104060$ $36096350$ $13240594400$ $4794339367136$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-1$ $45$ $305$ $2297$ $16974$ $118185$ $821547$ $5757537$ $40346945$ $282434600$

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 endomorphism algebra of this simple isogeny class is 6.0.30088184.1.

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.j_bm_ei$2$(not in LMFDB)