Properties

Label 2.11.ak_bu
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 yes

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Invariants

Base field:  $\F_{11}$
Dimension:  $2$
L-polynomial:  $( 1 - 6 x + 11 x^{2} )( 1 - 4 x + 11 x^{2} )$
  $1 - 10 x + 46 x^{2} - 110 x^{3} + 121 x^{4}$
Frobenius angles:  $\pm0.140218899004$, $\pm0.293962833700$
Angle rank:  $2$ (numerical)
Jacobians:  $2$
Isomorphism classes:  8

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

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $48$ $13824$ $1839600$ $218087424$ $26026361328$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $2$ $114$ $1382$ $14894$ $161602$ $1771938$ $19487302$ $214372894$ $2358056642$ $25937838354$

Jacobians and polarizations

This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:

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.ae 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.ac_ac$2$2.121.ai_gc
2.11.c_ac$2$2.121.ai_gc
2.11.k_bu$2$2.121.ai_gc