Properties

Label 3.11.an_dd_amk
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:  $( 1 - x + 11 x^{2} )( 1 - 6 x + 11 x^{2} )^{2}$
  $1 - 13 x + 81 x^{2} - 322 x^{3} + 891 x^{4} - 1573 x^{5} + 1331 x^{6}$
Frobenius angles:  $\pm0.140218899004$, $\pm0.140218899004$, $\pm0.451829325548$
Angle rank:  $2$ (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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $396$ $1667952$ $2355076944$ $3115894459392$ $4187575203146676$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-1$ $115$ $1328$ $14535$ $161449$ $1777876$ $19512499$ $214402895$ $2357984768$ $25937637955$

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 2 $\times$ 1.11.ab 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 are some of the twists of this isogeny class.

TwistExtension degreeCommon base change
3.11.al_cf_ahy$2$(not in LMFDB)
3.11.ab_ad_o$2$(not in LMFDB)
3.11.b_ad_ao$2$(not in LMFDB)
3.11.l_cf_hy$2$(not in LMFDB)
3.11.n_dd_mk$2$(not in LMFDB)
3.11.f_be_ed$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
3.11.al_cf_ahy$2$(not in LMFDB)
3.11.ab_ad_o$2$(not in LMFDB)
3.11.b_ad_ao$2$(not in LMFDB)
3.11.l_cf_hy$2$(not in LMFDB)
3.11.n_dd_mk$2$(not in LMFDB)
3.11.f_be_ed$3$(not in LMFDB)
3.11.ab_z_ao$4$(not in LMFDB)
3.11.b_z_o$4$(not in LMFDB)
3.11.ah_bq_agb$6$(not in LMFDB)
3.11.af_be_aed$6$(not in LMFDB)
3.11.h_bq_gb$6$(not in LMFDB)
3.11.af_x_ads$8$(not in LMFDB)
3.11.ad_p_adc$8$(not in LMFDB)
3.11.d_p_dc$8$(not in LMFDB)
3.11.f_x_ds$8$(not in LMFDB)