Properties

Label 3.11.aq_en_atc
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:  $( 1 - 4 x + 11 x^{2} )( 1 - 6 x + 11 x^{2} )^{2}$
Frobenius angles:  $\pm0.140218899004$, $\pm0.140218899004$, $\pm0.293962833700$
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
$A(\F_{q^r})$ 288 1492992 2417234400 3203268083712 4203934039866528 5568437853216000000 7403646096458768439648 9851624936637438269325312 13110967172879088971994093600 17449746167746488620161164183552

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -4 100 1364 14940 162076 1774276 19496116 214400060 2358122684 25937935780

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
The isogeny class factors as 1.11.ag 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.ai_v_abg$2$(not in LMFDB)
3.11.ae_ad_ce$2$(not in LMFDB)
3.11.e_ad_ace$2$(not in LMFDB)
3.11.i_v_bg$2$(not in LMFDB)
3.11.q_en_tc$2$(not in LMFDB)
3.11.c_m_bg$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.11.ai_v_abg$2$(not in LMFDB)
3.11.ae_ad_ce$2$(not in LMFDB)
3.11.e_ad_ace$2$(not in LMFDB)
3.11.i_v_bg$2$(not in LMFDB)
3.11.q_en_tc$2$(not in LMFDB)
3.11.c_m_bg$3$(not in LMFDB)
3.11.ae_z_ace$4$(not in LMFDB)
3.11.e_z_ce$4$(not in LMFDB)
3.11.ak_ci_aiy$6$(not in LMFDB)
3.11.ac_m_abg$6$(not in LMFDB)
3.11.k_ci_iy$6$(not in LMFDB)
3.11.ai_bj_aeq$8$(not in LMFDB)
3.11.a_d_ace$8$(not in LMFDB)
3.11.a_d_ce$8$(not in LMFDB)
3.11.i_bj_eq$8$(not in LMFDB)