Properties

Label 3.11.ao_dt_apo
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:  $( 1 - 6 x + 11 x^{2} )( 1 - 4 x + 11 x^{2} )^{2}$
  $1 - 14 x + 97 x^{2} - 404 x^{3} + 1067 x^{4} - 1694 x^{5} + 1331 x^{6}$
Frobenius angles:  $\pm0.140218899004$, $\pm0.293962833700$, $\pm0.293962833700$
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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $384$ $1769472$ $2575440000$ $3238162071552$ $4193575548057984$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $-2$ $120$ $1450$ $15100$ $161678$ $1769976$ $19478618$ $214359740$ $2358099550$ $25938154680$

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 $\times$ 1.11.ae 2 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.ag_r_abk$2$(not in LMFDB)
3.11.ac_b_ca$2$(not in LMFDB)
3.11.c_b_aca$2$(not in LMFDB)
3.11.g_r_bk$2$(not in LMFDB)
3.11.o_dt_po$2$(not in LMFDB)
3.11.ac_ai_cg$3$(not in LMFDB)

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

TwistExtension degreeCommon base change
3.11.ag_r_abk$2$(not in LMFDB)
3.11.ac_b_ca$2$(not in LMFDB)
3.11.c_b_aca$2$(not in LMFDB)
3.11.g_r_bk$2$(not in LMFDB)
3.11.o_dt_po$2$(not in LMFDB)
3.11.ac_ai_cg$3$(not in LMFDB)
3.11.ag_f_bk$4$(not in LMFDB)
3.11.g_f_abk$4$(not in LMFDB)
3.11.ak_bo_aeo$6$(not in LMFDB)
3.11.c_ai_acg$6$(not in LMFDB)
3.11.k_bo_eo$6$(not in LMFDB)