# Properties

 Label 3.11.ao_dp_aoq 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: $( 1 - 2 x + 11 x^{2} )( 1 - 6 x + 11 x^{2} )^{2}$ Frobenius angles: $\pm0.140218899004$, $\pm0.140218899004$, $\pm0.402508885479$ 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 $A(\F_{q^r})$ 360 1632960 2399968440 3141135728640 4181026450329000 5572402721940600000 7409766361514515019640 9853266309353470175477760 13110525746317059899073494760 17449317334841144616690201144000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -2 112 1354 14652 161198 1775536 19512218 214435772 2358043294 25937298352

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
 The isogeny class factors as 1.11.ag 2 $\times$ 1.11.ac 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 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-2})$$$)$ 1.11.ac : $$\Q(\sqrt{-10})$$.
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.ak_bt_afs $2$ (not in LMFDB) 3.11.ac_ad_bc $2$ (not in LMFDB) 3.11.c_ad_abc $2$ (not in LMFDB) 3.11.k_bt_fs $2$ (not in LMFDB) 3.11.o_dp_oq $2$ (not in LMFDB) 3.11.e_y_de $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.11.ak_bt_afs $2$ (not in LMFDB) 3.11.ac_ad_bc $2$ (not in LMFDB) 3.11.c_ad_abc $2$ (not in LMFDB) 3.11.k_bt_fs $2$ (not in LMFDB) 3.11.o_dp_oq $2$ (not in LMFDB) 3.11.e_y_de $3$ (not in LMFDB) 3.11.ac_z_abc $4$ (not in LMFDB) 3.11.c_z_bc $4$ (not in LMFDB) 3.11.ai_bw_aha $6$ (not in LMFDB) 3.11.ae_y_ade $6$ (not in LMFDB) 3.11.i_bw_ha $6$ (not in LMFDB) 3.11.ag_bb_aea $8$ (not in LMFDB) 3.11.ac_l_acu $8$ (not in LMFDB) 3.11.c_l_cu $8$ (not in LMFDB) 3.11.g_bb_ea $8$ (not in LMFDB)