padic QpLC.random_element is broken

[mwageringel]

sage -t --long --random-seed=3113 src/sage/rings/padics/padic_base_leaves.py
**********************************************************************
File "src/sage/rings/padics/padic_base_leaves.py", line 1071, in sage.rings.padics.padic_base_leaves.pAdicFieldLattice.random_element
Failed example:
    K.random_element()   # random
Exception raised:
    Traceback (most recent call last):
      File "/usr/lib/python3.9/site-packages/sage/doctest/forker.py", line 714, in _run
        self.compile_and_execute(example, compiler, test.globs)
      File "/usr/lib/python3.9/site-packages/sage/doctest/forker.py", line 1133, in compile_and_execute
        exec(compiled, globs)
      File "<doctest sage.rings.padics.padic_base_leaves.pAdicFieldLattice.random_element[1]>", line 1, in <module>
        K.random_element()   # random
      File "/usr/lib/python3.9/site-packages/sage/rings/padics/padic_base_leaves.py", line 1093, in random_element
        x = ZZ.random_element(p**prec)
      File "sage/rings/integer_ring.pyx", line 718, in sage.rings.integer_ring.IntegerRing_class.random_element (build/cythonized/sage/rings/integer_ring.c:6533)
        self._randomize_mpz(z.value, x, y, distribution)
      File "sage/rings/integer_ring.pyx", line 749, in sage.rings.integer_ring.IntegerRing_class._randomize_mpz (build/cythonized/sage/rings/integer_ring.c:6764)
        n_max = x if isinstance(x, integer.Integer) else self(x)
      File "sage/structure/parent.pyx", line 898, in sage.structure.parent.Parent.__call__ (build/cythonized/sage/structure/parent.c:9341)
        return mor._call_(x)
      File "sage/rings/rational.pyx", line 4164, in sage.rings.rational.Q_to_Z._call_ (build/cythonized/sage/rings/rational.cpp:32503)
        raise TypeError("no conversion of this rational to integer")
    TypeError: no conversion of this rational to integer

where K = QpLC(2). Other possible seeds:

sage -t --long --random-seed=3098 src/sage/rings/padics/padic_base_leaves.py  # 1 doctest failed
sage -t --long --random-seed=3083 src/sage/rings/padics/padic_base_leaves.py  # 1 doctest failed

In #29979, a doctest was marked not tested because of this.

CC: @kliem @roed314

Component: padics

Issue created by migration from https://trac.sagemath.org/ticket/32126

[mwageringel]

comment:1

The offending code is

        if integral:
            val = 0
        else:
            val = ZZ.random_element()
        if prec is None:
            prec = self._prec_cap_absolute - val
        p = self.prime()
        x = ZZ.random_element(p**prec)

which is problematic as prec can be negative, so that ZZ.random_element gets called with a non-integral value. I am not sure what the intention is.