Properties

Label 3.11.ap_ea_aqq
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:  $( 1 - 6 x + 11 x^{2} )( 1 - 9 x + 39 x^{2} - 99 x^{3} + 121 x^{4} )$
Frobenius angles:  $\pm0.100899808413$, $\pm0.140218899004$, $\pm0.366706655625$
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
$A(\F_{q^r})$ 318 1539756 2374722558 3138798765024 4171825565612448 5563216585745102700 7406323568281759240038 9853404054619142247744384 13111195936794203960314102422 17449575838115732438710086890496

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -3 105 1341 14641 160842 1772613 19503159 214438769 2358163827 25937682600

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
The isogeny class factors as 1.11.ag $\times$ 2.11.aj_bn 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.ad_ae_bk$2$(not in LMFDB)
3.11.d_ae_abk$2$(not in LMFDB)
3.11.p_ea_qq$2$(not in LMFDB)