Properties

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

Learn more about

Invariants

Base field:  $\F_{11}$
Dimension:  $3$
L-polynomial:  $( 1 - 6 x + 11 x^{2} )( 1 - 7 x + 26 x^{2} - 77 x^{3} + 121 x^{4} )$
Frobenius angles:  $\pm0.0895839137776$, $\pm0.140218899004$, $\pm0.469832509767$
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
$A(\F_{q^r})$ 384 1603584 2281019904 3078522077184 4176699208201344 5575298046411571200 7406968087636782820224 9850850807949927939833856 13110397656412088165171129856 17449799457834088098969972455424

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 111 1286 14359 161029 1776456 19504855 214383215 2358020258 25938014991

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
The isogeny class factors as 1.11.ag $\times$ 2.11.ah_ba 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_af_c$2$(not in LMFDB)
3.11.b_af_ac$2$(not in LMFDB)
3.11.n_db_ly$2$(not in LMFDB)