Properties

Label 3.11.ao_du_aps
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:  $( 1 - 4 x + 11 x^{2} )( 1 - 5 x + 11 x^{2} )^{2}$
Frobenius angles:  $\pm0.228229222880$, $\pm0.228229222880$, $\pm0.293962833700$
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
$A(\F_{q^r})$ 392 1812608 2635337600 3285352000000 4217009424478312 5560429767977369600 7394930309112088549912 9846830536158121632000000 13109168384776327428357430400 17449341236759812446518301035648

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -2 122 1480 15314 162578 1771724 19473158 214295714 2357799160 25937333882

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
The isogeny class factors as 1.11.af 2 $\times$ 1.11.ae 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_s_abg$2$(not in LMFDB)
3.11.ae_i_m$2$(not in LMFDB)
3.11.e_i_am$2$(not in LMFDB)
3.11.g_s_bg$2$(not in LMFDB)
3.11.o_du_ps$2$(not in LMFDB)
3.11.b_f_cc$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.11.ag_s_abg$2$(not in LMFDB)
3.11.ae_i_m$2$(not in LMFDB)
3.11.e_i_am$2$(not in LMFDB)
3.11.g_s_bg$2$(not in LMFDB)
3.11.o_du_ps$2$(not in LMFDB)
3.11.b_f_cc$3$(not in LMFDB)
3.11.ae_o_am$4$(not in LMFDB)
3.11.e_o_m$4$(not in LMFDB)
3.11.aj_bt_agk$6$(not in LMFDB)
3.11.ab_f_acc$6$(not in LMFDB)
3.11.j_bt_gk$6$(not in LMFDB)