Properties

Label 3.11.an_df_amy
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:  $( 1 - 5 x + 11 x^{2} )( 1 - 8 x + 32 x^{2} - 88 x^{3} + 121 x^{4} )$
Frobenius angles:  $\pm0.0750991438595$, $\pm0.228229222880$, $\pm0.424900856141$
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
$A(\F_{q^r})$ 406 1725500 2416006936 3127468750000 4168862454145466 5563259443298312000 7402850858960619757646 9849421608098688000000000 13109285827450483843880118376 17449265349453557126655824637500

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 119 1364 14591 160729 1772624 19494019 214352111 2357820284 25937221079

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
The isogeny class factors as 1.11.af $\times$ 2.11.ai_bg and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{11}$
The base change of $A$ to $\F_{11^{4}}$ is 1.14641.afm 2 $\times$ 1.14641.iz. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{11^{4}}$.
Remainder of endomorphism lattice by field

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.ad_d_aq$2$(not in LMFDB)
3.11.d_d_q$2$(not in LMFDB)
3.11.n_df_my$2$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.11.ad_d_aq$2$(not in LMFDB)
3.11.d_d_q$2$(not in LMFDB)
3.11.n_df_my$2$(not in LMFDB)
3.11.d_d_q$4$(not in LMFDB)
3.11.af_b_by$8$(not in LMFDB)
3.11.af_v_aby$8$(not in LMFDB)
3.11.f_b_aby$8$(not in LMFDB)
3.11.f_v_by$8$(not in LMFDB)
3.11.al_cm_ajn$24$(not in LMFDB)
3.11.ab_e_ar$24$(not in LMFDB)
3.11.b_e_r$24$(not in LMFDB)
3.11.l_cm_jn$24$(not in LMFDB)