Properties

Label 3.7.ak_cb_agr
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:  $( 1 - 3 x + 7 x^{2} )( 1 - 7 x + 25 x^{2} - 49 x^{3} + 49 x^{4} )$
Frobenius angles:  $\pm0.162349854003$, $\pm0.308124534521$, $\pm0.351370772325$
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
$A(\F_{q^r})$ 95 136895 49521980 14716896975 4746304811600 1620469520571440 558424991759101205 191790410661932100975 65748600161030690773580 22540805470947977630841600

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

Decomposition and endomorphism algebra

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:
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.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)