# Properties

 Label 3.11.ao_du_aps 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: $( 1 - 4 x + 11 x^{2} )( 1 - 5 x + 11 x^{2} )^{2}$ Frobenius angles: $\pm0.228229222880$, $\pm0.228229222880$, $\pm0.293962833700$ 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 $A(\F_{q^r})$ 392 1812608 2635337600 3285352000000 4217009424478312 5560429767977369600 7394930309112088549912 9846830536158121632000000 13109168384776327428357430400 17449341236759812446518301035648

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -2 122 1480 15314 162578 1771724 19473158 214295714 2357799160 25937333882

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
 The isogeny class factors as 1.11.af 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: 1.11.af 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-19})$$$)$ 1.11.ae : $$\Q(\sqrt{-7})$$.
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_s_abg $2$ (not in LMFDB) 3.11.ae_i_m $2$ (not in LMFDB) 3.11.e_i_am $2$ (not in LMFDB) 3.11.g_s_bg $2$ (not in LMFDB) 3.11.o_du_ps $2$ (not in LMFDB) 3.11.b_f_cc $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.11.ag_s_abg $2$ (not in LMFDB) 3.11.ae_i_m $2$ (not in LMFDB) 3.11.e_i_am $2$ (not in LMFDB) 3.11.g_s_bg $2$ (not in LMFDB) 3.11.o_du_ps $2$ (not in LMFDB) 3.11.b_f_cc $3$ (not in LMFDB) 3.11.ae_o_am $4$ (not in LMFDB) 3.11.e_o_m $4$ (not in LMFDB) 3.11.aj_bt_agk $6$ (not in LMFDB) 3.11.ab_f_acc $6$ (not in LMFDB) 3.11.j_bt_gk $6$ (not in LMFDB)