python3.10 with numpy - ModuleNotFoundError: No module named 'numpy.core._multiarray_umath'

I'm trying to install numpy for python3.10 (not in a venv). I installed python 3.10 using

sudo apt install python3.10

pip didn't seem to be installed so I did

jeremy@jeremy-Blade:/$ python3.10 -m pip install numpy
jeremy@jeremy-Blade:/$ wget https://bootstrap.pypa.io/get-pip.py
jeremy@jeremy-Blade:/$ python3.10 get-pip.py
jeremy@jeremy-Blade:/$ python3.10 -m pip --version 
pip 21.3.1 from /home/jeremy/.local/lib/python3.10/site-packages/pip (python 3.10)

and then tried

jeremy@jeremy-Blade:/$ python3.10 -m pip install numpy
Defaulting to user installation because normal site-packages is not writeable
Requirement already satisfied: numpy in /usr/lib/python3/dist-packages (1.17.4)

But when I try to import numpy :

Python 3.10.0 (default, Oct  4 2021, 22:09:55) [GCC 9.3.0] on linux
Type "help", "copyright", "credits" or "license" for more information.
>>> import numpy 
Traceback (most recent call last):
  File "/usr/lib/python3/dist-packages/numpy/core/__init__.py", line 17, in <module>
    from . import multiarray
  File "/usr/lib/python3/dist-packages/numpy/core/multiarray.py", line 14, in <module>
    from . import overrides
  File "/usr/lib/python3/dist-packages/numpy/core/overrides.py", line 7, in <module>
    from numpy.core._multiarray_umath import (
ModuleNotFoundError: No module named 'numpy.core._multiarray_umath'

During handling of the above exception, another exception occurred:

Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "/usr/lib/python3/dist-packages/numpy/__init__.py", line 142, in <module>
    from . import core
  File "/usr/lib/python3/dist-packages/numpy/core/__init__.py", line 47, in <module>
    raise ImportError(msg)
ImportError: 

IMPORTANT: PLEASE READ THIS FOR ADVICE ON HOW TO SOLVE THIS ISSUE!

Importing the numpy c-extensions failed.
- Try uninstalling and reinstalling numpy.
- If you have already done that, then:
  1. Check that you expected to use Python3.10 from "/usr/bin/python3.10",
     and that you have no directories in your PATH or PYTHONPATH that can
     interfere with the Python and numpy version "1.17.4" you're trying to use.
  2. If (1) looks fine, you can open a new issue at
     https://github.com/numpy/numpy/issues.  Please include details on:
     - how you installed Python
     - how you installed numpy
     - your operating system
     - whether or not you have multiple versions of Python installed
     - if you built from source, your compiler versions and ideally a build log
...
Original error was: No module named 'numpy.core._multiarray_umath'

my path is

jeremy@jeremy-Blade:/$env|grep PATH
LD_LIBRARY_PATH=/usr/local/cuda-10.2/targets/x86_64-linux/lib/stubs
PATH=/home/jeremy/.local/bin:/usr/local/cuda-10.2/bin:/usr/local/cuda-10.2/targets/x86_64-linux/lib/stubs:/usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin:/usr/games:/usr/local/games:/snap/bin

(so PYTHONPATH is not set - if I do set it to /usr/bin/python3.10 this doesn't seem to change anything) , and following the recommendations in the error message I don't seem to see any problem here. (I uninstalled and reinstalled numpy for python3.10, no change seen in problem) My OS is ubuntu 20.04. Sorry if this is something obvious or unrelated to numpy.