Corrupted polynomial (wrong results/exceptions, crash SIGABRT) obtained from sparse FractionField (FLINT)

[hellman]

Steps To Reproduce

The following code creates a univariate polynomial $f(x)$ modulo 37, which has very strange and wrong behavior. It misses some attributes (e.g. factor(f) can not find .modulus, f.radical() produces wrong result, wrong arithmetic, probably more).

print(version())  
x = PolynomialRing(GF(37), ['x'], sparse=True).fraction_field().gen()
f = (x^8 + 16*x^6 + 4*x^4 + x^2 + 12).numerator()
print(type(f), f.parent())
print("  f", f)
print("f+1", f+1)
print("rad", f.radical())
print("r^2", f.radical()**2)
print(factor(f))

Output:

SageMath version 10.2, Release Date: 2023-12-03
<class 'sage.rings.polynomial.polynomial_zmod_flint.Polynomial_zmod_flint'> Sparse Univariate Polynomial Ring in x over Finite Field of size 37
  f x^8 + 16*x^6 + 4*x^4 + x^2 + 12
f+1 x^8 + 16*x^6 + 4*x^4 + x^2 + 12
rad x^4 + 15*x^2 + 26
r^2 x^8 + x^4
...
AttributeError: 'PolynomialRing_field_with_category' object has no attribute 'modulus'

While the actual radical should be just the square root of the polynomial which is:

sage: x = GF(37)['x'].gen()
sage: f = (x^8 + 16*x^6 + 4*x^4 + x^2 + 12)
sage: f.radical()
x^4 + 8*x^2 + 7
sage: f.radical()**2
x^8 + 16*x^6 + 4*x^4 + x^2 + 12
sage: factor(f)
(x + 6)^2 * (x + 17)^2 * (x + 20)^2 * (x + 31)^2

Expected Behavior

The output object is well formed and performs correctly (e.g. factor(f) succeeds, f.radical() is computed correctly, f+1 is computed correctly, etc.).

Actual Behavior

The output object is corrupted and yields wrong results / exceptions.

Additional Information

It might be that flint does not support sparse polynomials, while Sage wrongfully forces a dense object to be interpreted as a sparse one?

sage: PolynomialRing(GF(37), ['x'], sparse=False, implementation="FLINT")
Univariate Polynomial Ring in x over Finite Field of size 37
sage: PolynomialRing(GF(37), ['x'], sparse=True, implementation="FLINT")
---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
...
ValueError: unknown implementation 'FLINT' for sparse polynomial rings over Finite Field of size 37

Environment

- **OS**: Linux Mint 21.3
- **Sage Version**: 10.2

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

[hellman]

Actually, f-1 causes a crash.

sage: print("f-1", f-1)
Exception (FLINT memory_manager). Unable to allocate memory (3999070376720).

[hellman]

May be related to other FLINT crashes #37317 #37169

(But this seems to be an actual memory corruption rather than an unhandled FLINT exception, also here the modulus is prime)