Properties

Label 3.11.ao_ds_apk
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:  $( 1 - 4 x + 11 x^{2} )( 1 - 10 x + 45 x^{2} - 110 x^{3} + 121 x^{4} )$
Frobenius angles:  $\pm0.0820279942768$, $\pm0.293962833700$, $\pm0.318205720493$
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
$A(\F_{q^r})$ 376 1726592 2515928800 3190334540288 4167627913709656 5545555836388198400 7395195612291732009896 9849954804418201616640000 13111091587340037275848679200 17449980355195646168542008874112

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -2 118 1420 14882 160678 1766980 19473858 214363714 2358145060 25938283878

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
The isogeny class factors as 1.11.ae $\times$ 2.11.ak_bt 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.ag_q_abo$2$(not in LMFDB)
3.11.g_q_bo$2$(not in LMFDB)
3.11.o_ds_pk$2$(not in LMFDB)