Properties

Label 3.7.ak_cb_agr
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:  $( 1 - 3 x + 7 x^{2} )( 1 - 7 x + 25 x^{2} - 49 x^{3} + 49 x^{4} )$
  $1 - 10 x + 53 x^{2} - 173 x^{3} + 371 x^{4} - 490 x^{5} + 343 x^{6}$
Frobenius angles:  $\pm0.162349854003$, $\pm0.308124534521$, $\pm0.351370772325$
Angle rank:  $3$ (numerical)
Isomorphism classes:  1

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $95$ $136895$ $49521980$ $14716896975$ $4746304811600$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-2$ $56$ $415$ $2548$ $16803$ $117077$ $823366$ $5771092$ $40375855$ $282493611$

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.ad $\times$ 2.7.ah_z 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 is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
3.7.ae_l_ax$2$(not in LMFDB)
3.7.e_l_x$2$(not in LMFDB)
3.7.k_cb_gr$2$(not in LMFDB)