Properties

Label 3.11.aq_en_atd
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:  $( 1 - 5 x + 11 x^{2} )( 1 - 11 x + 51 x^{2} - 121 x^{3} + 121 x^{4} )$
Frobenius angles:  $\pm0.0215640055172$, $\pm0.228229222880$, $\pm0.270299311731$
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
$A(\F_{q^r})$ 287 1488095 2408204372 3188429549375 4185769049802032 5552464513772202320 7393096432039702835437 9846257309200912727019375 13108906041999147831913733012 17449225520380929509044580000000

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -4 100 1361 14876 161381 1769185 19468326 214283236 2357751971 25937161875

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
The isogeny class factors as 1.11.af $\times$ 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.ag_h_n$2$(not in LMFDB)
3.11.g_h_an$2$(not in LMFDB)
3.11.q_en_td$2$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.11.ag_h_n$2$(not in LMFDB)
3.11.g_h_an$2$(not in LMFDB)
3.11.q_en_td$2$(not in LMFDB)
3.11.ag_h_x$5$(not in LMFDB)
3.11.ag_bl_aex$5$(not in LMFDB)
3.11.ab_ad_cg$5$(not in LMFDB)
3.11.e_h_ah$5$(not in LMFDB)
3.11.ao_dt_apn$10$(not in LMFDB)
3.11.aj_bl_aeo$10$(not in LMFDB)
3.11.ae_ad_cp$10$(not in LMFDB)
3.11.ae_h_h$10$(not in LMFDB)
3.11.ae_bb_adf$10$(not in LMFDB)
3.11.b_ad_acg$10$(not in LMFDB)
3.11.e_ad_acp$10$(not in LMFDB)
3.11.e_bb_df$10$(not in LMFDB)
3.11.g_h_ax$10$(not in LMFDB)
3.11.g_bl_ex$10$(not in LMFDB)
3.11.j_bl_eo$10$(not in LMFDB)
3.11.o_dt_pn$10$(not in LMFDB)