`sqrt(Mod(,),all=True)` should raise more informative `NotImplmentedError`

[maxale]

Steps To Reproduce

Check this out:

sage: is_square(Mod(2, 101010))
False
sage: sqrt(Mod(2, 101010),all=True)
---------------------------------------------------------------------------
NotImplementedError                       Traceback (most recent call last)

Expected Behavior

Sage should simply return [], ie. an empty list of roots (as there are none exist). In fact, a reason (which I believe is tested by is_squre()) is very simple: the modulus 101010 is divisible by 3, while 2 is not a square modulo 3.

Actual Behavior

Raises NotImplemented, while is_square() easily establishes that there are no solutions.

Additional Information

Btw, sqrt(Mod(2, 101010)) (ie. without all=True) returns a weird/incorrect result, which I submit as another issue #38803.

Environment

• OS: Ubuntu 24.04.1 LTS
• Sage Version: 10.5.beta6

Checklist

• I have searched the existing issues for a bug report that matches the one I want to file, without success.
• I have read the documentation and troubleshoot guide

[maxale]

As suggested in the discussion of #38803, the requested behavior can be obtained with parameter extend=False:

sage: sqrt(Mod(2, 101010),all=True,extend=False)
[]

The question is then whether extend=True should really be the default.

[maxale]

While it's NotImplemented, should the error message suggest to use extend=False? Example from polynomial roots:

sage: x = polygen(Zmod(101010))
sage: (x^2 - 2).roots()
[...]
NotImplementedError: root finding with multiplicities for this polynomial not implemented (try the multiplicities=False option)

[yyyyx4]

(I've removed the "bug" label since some part of this is documented behaviour, and the other part is just a lack of functionality, which isn't usually considered a bug.)