# Properties

 Label 3.11.ao_ds_api 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: $( 1 - 6 x + 11 x^{2} )( 1 - 5 x + 11 x^{2} )( 1 - 3 x + 11 x^{2} )$ Frobenius angles: $\pm0.140218899004$, $\pm0.228229222880$, $\pm0.350615407277$ Angle rank: $3$ (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 $A(\F_{q^r})$ 378 1735020 2531142432 3214992060000 4193507513145258 5562671473666944000 7402105932333456681918 9850932518460791934960000 13110305954076126290290434912 17449326704437058315563297195500

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -2 118 1426 14994 161678 1772440 19492058 214384994 2358003766 25937312278

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
 The isogeny class factors as 1.11.ag $\times$ 1.11.af $\times$ 1.11.ad and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
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.ai_be_adi $2$ (not in LMFDB) 3.11.ae_g_c $2$ (not in LMFDB) 3.11.ac_a_bu $2$ (not in LMFDB) 3.11.c_a_abu $2$ (not in LMFDB) 3.11.e_g_ac $2$ (not in LMFDB) 3.11.i_be_di $2$ (not in LMFDB) 3.11.o_ds_pi $2$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.11.ai_be_adi $2$ (not in LMFDB) 3.11.ae_g_c $2$ (not in LMFDB) 3.11.ac_a_bu $2$ (not in LMFDB) 3.11.c_a_abu $2$ (not in LMFDB) 3.11.e_g_ac $2$ (not in LMFDB) 3.11.i_be_di $2$ (not in LMFDB) 3.11.o_ds_pi $2$ (not in LMFDB)