Properties

Label 3.11.ao_dt_apo
Base Field $\F_{11}$
Dimension $3$
Ordinary Yes
$p$-rank $3$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{11}$
Dimension:  $3$
L-polynomial:  $( 1 - 6 x + 11 x^{2} )( 1 - 4 x + 11 x^{2} )^{2}$
Frobenius angles:  $\pm0.140218899004$, $\pm0.293962833700$, $\pm0.293962833700$
Angle rank:  $2$ (numerical)

This isogeny class is not simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $3$
Slopes:  $[0, 0, 0, 1, 1, 1]$

Point counts

This isogeny class is principally polarizable, but does not contain a Jacobian.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 384 1769472 2575440000 3238162071552 4193575548057984 5554939751424000000 7397001190199816697984 9849772122421444038623232 13110838543808696379936240000 17449893435069558118523922481152

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

Decomposition and endomorphism algebra

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:
All geometric endomorphisms are defined over $\F_{11}$.

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)