# Properties

 Label 3.11.aq_eo_ati 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: $( 1 - 6 x + 11 x^{2} )( 1 - 5 x + 11 x^{2} )^{2}$ Frobenius angles: $\pm0.140218899004$, $\pm0.228229222880$, $\pm0.228229222880$ 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 $A(\F_{q^r})$ 294 1529388 2473452576 3249949500000 4227425799974454 5573941210112486400 7401573355048140841314 9848682797040213462000000 13109296997460963836814159456 17449193974096987274709084680748

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -4 102 1394 15154 162976 1776024 19490656 214336034 2357822294 25937114982

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
 The isogeny class factors as 1.11.ag $\times$ 1.11.af 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.af 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-19})$$$)$
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_i_s $2$ (not in LMFDB) 3.11.ae_ac_ck $2$ (not in LMFDB) 3.11.e_ac_ack $2$ (not in LMFDB) 3.11.g_i_as $2$ (not in LMFDB) 3.11.q_eo_ti $2$ (not in LMFDB) 3.11.ab_af_ba $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.11.ag_i_s $2$ (not in LMFDB) 3.11.ae_ac_ck $2$ (not in LMFDB) 3.11.e_ac_ack $2$ (not in LMFDB) 3.11.g_i_as $2$ (not in LMFDB) 3.11.q_eo_ti $2$ (not in LMFDB) 3.11.ab_af_ba $3$ (not in LMFDB) 3.11.ag_o_as $4$ (not in LMFDB) 3.11.g_o_s $4$ (not in LMFDB) 3.11.al_cd_ahm $6$ (not in LMFDB) 3.11.b_af_aba $6$ (not in LMFDB) 3.11.l_cd_hm $6$ (not in LMFDB)