Properties

Label 2.11.al_ca
Base field $\F_{11}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian no

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Invariants

Base field:  $\F_{11}$
Dimension:  $2$
L-polynomial:  $( 1 - 6 x + 11 x^{2} )( 1 - 5 x + 11 x^{2} )$
  $1 - 11 x + 52 x^{2} - 121 x^{3} + 121 x^{4}$
Frobenius angles:  $\pm0.140218899004$, $\pm0.228229222880$
Angle rank:  $2$ (numerical)
Jacobians:  $0$
Isomorphism classes:  1

This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $2$
Slopes:  $[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $42$ $12852$ $1802808$ $218484000$ $26131191702$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $1$ $105$ $1354$ $14921$ $162251$ $1774962$ $19493321$ $214361041$ $2357918014$ $25937318505$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{11}$.

Endomorphism algebra over $\F_{11}$
The isogeny class factors as 1.11.ag $\times$ 1.11.af and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
2.11.ab_ai$2$2.121.ar_ky
2.11.b_ai$2$2.121.ar_ky
2.11.l_ca$2$2.121.ar_ky