Properties

Label 3.11.an_da_als
Base Field $\F_{11}$
Dimension $3$
Ordinary Yes
$p$-rank $3$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{11}$
Dimension:  $3$
L-polynomial:  $( 1 - 6 x + 11 x^{2} )( 1 - 7 x + 25 x^{2} - 77 x^{3} + 121 x^{4} )$
Frobenius angles:  $\pm0.0530380253560$, $\pm0.140218899004$, $\pm0.477974681599$
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
$A(\F_{q^r})$ 378 1571724 2244302802 3058612625376 4168520769496608 5570391311448648300 7403839930515044019402 9849519412573842587988864 13110008037839783389644043698 17449655321335045828037617234944

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 109 1265 14265 160714 1774897 19496623 214354241 2357950181 25937800744

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
The isogeny class factors as 1.11.ag $\times$ 2.11.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 $\F_{11}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.11.ab_ag_ae$2$(not in LMFDB)
3.11.b_ag_e$2$(not in LMFDB)
3.11.n_da_ls$2$(not in LMFDB)