Properties

Label 3.11.ap_ec_ard
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:  $( 1 - 5 x + 11 x^{2} )( 1 - 10 x + 45 x^{2} - 110 x^{3} + 121 x^{4} )$
  $1 - 15 x + 106 x^{2} - 445 x^{3} + 1166 x^{4} - 1815 x^{5} + 1331 x^{6}$
Frobenius angles:  $\pm0.0820279942768$, $\pm0.228229222880$, $\pm0.318205720493$
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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $329$ $1605191$ $2465610224$ $3196135929875$ $4184414508938279$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-3$ $109$ $1392$ $14909$ $161327$ $1770004$ $19479877$ $214351861$ $2358006432$ $25937764029$

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.af $\times$ 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:

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.af_g_f$2$(not in LMFDB)
3.11.f_g_af$2$(not in LMFDB)
3.11.p_ec_rd$2$(not in LMFDB)