Properties

Label 3.11.an_dg_ang
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:  $( 1 - 2 x + 11 x^{2} )( 1 - 11 x + 51 x^{2} - 121 x^{3} + 121 x^{4} )$
  $1 - 13 x + 84 x^{2} - 344 x^{3} + 924 x^{4} - 1573 x^{5} + 1331 x^{6}$
Frobenius angles:  $\pm0.0215640055172$, $\pm0.270299311731$, $\pm0.402508885479$
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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $410$ $1750700$ $2439798890$ $3120909864800$ $4146259914764000$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-1$ $121$ $1379$ $14561$ $159854$ $1767421$ $19478409$ $214330801$ $2357811209$ $25937044296$

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.ac $\times$ 2.11.al_bz 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.aj_bo_afk$2$(not in LMFDB)
3.11.j_bo_fk$2$(not in LMFDB)
3.11.n_dg_ng$2$(not in LMFDB)

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

TwistExtension degreeCommon base change
3.11.aj_bo_afk$2$(not in LMFDB)
3.11.j_bo_fk$2$(not in LMFDB)
3.11.n_dg_ng$2$(not in LMFDB)
3.11.ad_e_ae$5$(not in LMFDB)
3.11.ad_bi_acm$5$(not in LMFDB)
3.11.c_j_cy$5$(not in LMFDB)
3.11.h_bi_em$5$(not in LMFDB)
3.11.al_cs_aku$10$(not in LMFDB)
3.11.ah_bi_aem$10$(not in LMFDB)
3.11.ag_z_adw$10$(not in LMFDB)
3.11.ac_j_acy$10$(not in LMFDB)
3.11.ab_a_bo$10$(not in LMFDB)
3.11.ab_be_au$10$(not in LMFDB)
3.11.b_a_abo$10$(not in LMFDB)
3.11.b_be_u$10$(not in LMFDB)
3.11.d_e_e$10$(not in LMFDB)
3.11.d_bi_cm$10$(not in LMFDB)
3.11.g_z_dw$10$(not in LMFDB)
3.11.l_cs_ku$10$(not in LMFDB)