Properties

Label 3.11.an_dc_ame
Base field $\F_{11}$
Dimension $3$
$p$-rank $3$
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:  $3$
L-polynomial:  $( 1 - 6 x + 11 x^{2} )( 1 - 7 x + 27 x^{2} - 77 x^{3} + 121 x^{4} )$
  $1 - 13 x + 80 x^{2} - 316 x^{3} + 880 x^{4} - 1573 x^{5} + 1331 x^{6}$
Frobenius angles:  $\pm0.116678659763$, $\pm0.140218899004$, $\pm0.461158112795$
Angle rank:  $3$ (numerical)

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

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $3$
Slopes:  $[0, 0, 0, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $390$ $1635660$ $2317941990$ $3097612908000$ $4183049482716000$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-1$ $113$ $1307$ $14449$ $161274$ $1777445$ $19510049$ $214398289$ $2358023177$ $25937911528$

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 $\times$ 2.11.ah_bb 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
3.11.ab_ae_i$2$(not in LMFDB)
3.11.b_ae_ai$2$(not in LMFDB)
3.11.n_dc_me$2$(not in LMFDB)