Properties

Label 3.11.ao_dr_apf
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:  $( 1 - 3 x + 11 x^{2} )( 1 - 11 x + 51 x^{2} - 121 x^{3} + 121 x^{4} )$
Frobenius angles:  $\pm0.0215640055172$, $\pm0.270299311731$, $\pm0.350615407277$
Angle rank:  $3$ (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
$A(\F_{q^r})$ 369 1688175 2464372404 3154133836575 4152184991334864 5541238200229477200 7393628399371374078819 9848506476570552624992175 13109914968524435938233985524 17449358250960962865742500000000

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -2 116 1393 14716 160083 1765601 19469728 214332196 2357933443 25937359171

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
The isogeny class factors as 1.11.ad $\times$ 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:
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.ai_bd_adl$2$(not in LMFDB)
3.11.i_bd_dl$2$(not in LMFDB)
3.11.o_dr_pf$2$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.11.ai_bd_adl$2$(not in LMFDB)
3.11.i_bd_dl$2$(not in LMFDB)
3.11.o_dr_pf$2$(not in LMFDB)
3.11.ae_f_f$5$(not in LMFDB)
3.11.ae_bj_adh$5$(not in LMFDB)
3.11.b_f_cs$5$(not in LMFDB)
3.11.g_z_cx$5$(not in LMFDB)
3.11.am_db_amj$10$(not in LMFDB)
3.11.ah_bd_aec$10$(not in LMFDB)
3.11.ag_z_acx$10$(not in LMFDB)
3.11.ac_ab_bx$10$(not in LMFDB)
3.11.ac_bd_abp$10$(not in LMFDB)
3.11.ab_f_acs$10$(not in LMFDB)
3.11.c_ab_abx$10$(not in LMFDB)
3.11.c_bd_bp$10$(not in LMFDB)
3.11.e_f_af$10$(not in LMFDB)
3.11.e_bj_dh$10$(not in LMFDB)
3.11.h_bd_ec$10$(not in LMFDB)
3.11.m_db_mj$10$(not in LMFDB)