# 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

## 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.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 406 1725500 2416006936 3127468750000 4168862454145466 5563259443298312000 7402850858960619757646 9849421608098688000000000 13109285827450483843880118376 17449265349453557126655824637500

 $r$ 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: 1.14641.afm 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-6})$$$)$ 1.14641.iz : $$\Q(\sqrt{-19})$$.
All geometric endomorphisms are defined over $\F_{11^{4}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{11^{2}}$  The base change of $A$ to $\F_{11^{2}}$ is 1.121.ad $\times$ 2.121.a_afm. The endomorphism algebra for each factor is:

## Base change

This is a primitive isogeny class.

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common 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.
 Twist Extension Degree Common 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)