trac 33360: avoid factoring in is_prime()

In the added TEST, the ideal norm is product of two primes but factoring
this product takes about half an hour, so factoring the ideal is slow.

To fix the issue, we use PARI's idealismaximal() function instead.

Co-authored-by: Gonzalo Tornaría <[email protected]>
Co-authored-by: Lorenz Panny <[email protected]>

Author: tornaria

src/sage/rings/number_field/number_field_ideal.py

@@ -996,16 +996,38 @@ def is_prime(self):
             False
             sage: K.ideal(17).is_prime()  # ramified
             False
+
+        TESTS:
+
+        Check that we do not factor the norm of the ideal, this used
+        to take half an hour, see :trac:`33360`::
+
+            sage: K.<a,b,c> = NumberField([x^2-2,x^2-3,x^2-5])
+            sage: t = (((-2611940*c + 1925290/7653)*b - 1537130/7653*c
+            ....:       + 10130950)*a + (1343014/7653*c - 8349770)*b
+            ....:       + 6477058*c - 2801449990/4002519)
+            sage: t.is_prime()
+            False
         """
         try:
             return self._pari_prime is not None
         except AttributeError:
-            F = self.factor()  # factorization with caching
-            if len(F) != 1 or F[0][1] != 1:
-                self._pari_prime = None
-            else:
-                self._pari_prime = F[0][0]._pari_prime
-            return self._pari_prime is not None
+            pass
+
+        K = self.number_field().pari_nf()
+        I = self.pari_hnf()
+
+        candidate = K.idealismaximal(I) or None
+
+        # PARI uses probabilistic primality testing inside idealismaximal().
+        if get_flag(None, 'arithmetic'):
+            # proof required, check using isprime()
+            if candidate and not candidate[0].isprime():
+                candidate = None
+
+        self._pari_prime = candidate
+
+        return self._pari_prime is not None
 
     def pari_prime(self):
         r"""