# Properties

 Label 3.11.ao_dt_apo 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: $( 1 - 6 x + 11 x^{2} )( 1 - 4 x + 11 x^{2} )^{2}$ Frobenius angles: $\pm0.140218899004$, $\pm0.293962833700$, $\pm0.293962833700$ 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.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 384 1769472 2575440000 3238162071552 4193575548057984 5554939751424000000 7397001190199816697984 9849772122421444038623232 13110838543808696379936240000 17449893435069558118523922481152

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -2 120 1450 15100 161678 1769976 19478618 214359740 2358099550 25938154680

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
 The isogeny class factors as 1.11.ag $\times$ 1.11.ae 2 and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: 1.11.ag : $$\Q(\sqrt{-2})$$. 1.11.ae 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-7})$$$)$
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.
 Twist Extension Degree Common base change 3.11.ag_r_abk $2$ (not in LMFDB) 3.11.ac_b_ca $2$ (not in LMFDB) 3.11.c_b_aca $2$ (not in LMFDB) 3.11.g_r_bk $2$ (not in LMFDB) 3.11.o_dt_po $2$ (not in LMFDB) 3.11.ac_ai_cg $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.11.ag_r_abk $2$ (not in LMFDB) 3.11.ac_b_ca $2$ (not in LMFDB) 3.11.c_b_aca $2$ (not in LMFDB) 3.11.g_r_bk $2$ (not in LMFDB) 3.11.o_dt_po $2$ (not in LMFDB) 3.11.ac_ai_cg $3$ (not in LMFDB) 3.11.ag_f_bk $4$ (not in LMFDB) 3.11.g_f_abk $4$ (not in LMFDB) 3.11.ak_bo_aeo $6$ (not in LMFDB) 3.11.c_ai_acg $6$ (not in LMFDB) 3.11.k_bo_eo $6$ (not in LMFDB)