Properties

Label 3.11.as_fl_axo
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:  $( 1 - 6 x + 11 x^{2} )^{3}$
  $1 - 18 x + 141 x^{2} - 612 x^{3} + 1551 x^{4} - 2178 x^{5} + 1331 x^{6}$
Frobenius angles:  $\pm0.140218899004$, $\pm0.140218899004$, $\pm0.140218899004$
Angle rank:  $1$ (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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $216$ $1259712$ $2268747144$ $3168750108672$ $4214318118039576$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-6$ $80$ $1278$ $14780$ $162474$ $1778576$ $19513614$ $214440380$ $2358145818$ $25937716880$

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 3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-2}) \)$)$

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.ag_ad_dg$2$(not in LMFDB)
3.11.g_ad_adg$2$(not in LMFDB)
3.11.s_fl_xo$2$(not in LMFDB)
3.11.a_a_as$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
3.11.ag_ad_dg$2$(not in LMFDB)
3.11.g_ad_adg$2$(not in LMFDB)
3.11.s_fl_xo$2$(not in LMFDB)
3.11.a_a_as$3$(not in LMFDB)
3.11.ag_z_adg$4$(not in LMFDB)
3.11.g_z_dg$4$(not in LMFDB)
3.11.am_cu_akw$6$(not in LMFDB)
3.11.a_a_s$6$(not in LMFDB)
3.11.m_cu_kw$6$(not in LMFDB)
3.11.ak_br_afg$8$(not in LMFDB)
3.11.ac_af_bo$8$(not in LMFDB)
3.11.c_af_abo$8$(not in LMFDB)
3.11.k_br_fg$8$(not in LMFDB)