Add methods to deal with isogenies in FiniteDrinfeldModule

The added methods are:
    - is_morphism,
    - is_endomorphism,
    - is_automorphism,
    - is_isogeny,
    - velu.

src/sage/modules/finite_drinfeld_module.py

@@ -276,6 +276,30 @@ def is_supersingular(self):
     def is_ordinary(self):
         return not self.is_supersingular()
 
+    def is_morphism(self, candidate):
+        return candidate == 0 and self.is_isogeny(candidate)
+
+    def is_isogeny(self, candidate):
+        if not candidate in self.ore_polring():
+            raise TypeError('The candidate must be in the Ore polynomial ' \
+                    'ring')
+        if candidate == 0:
+            return False
+        elif candidate in self.ore_polring().base_ring():
+            return True
+        else:
+            return self.characteristic().degree().divides(candidate.valuation()) \
+                and candidate.right_divides(candidate * self.gen())
+
+    def is_endomorphism(self, candidate):
+        return candidate == 0 or self == self.velu(candidate)
+
+    def is_automorphism(self, candidate):
+        if not candidate in self.ore_polring():
+            raise TypeError('The candidate must be in the Ore polynomial ' \
+                    'ring')
+        return candidate != 0 and candidate in self._Fq()
+
     def j(self):
         if self.rank() != 2:
             raise ValueError('The j-invariant is only defined for rank 2 ' \
@@ -288,6 +312,26 @@ def j(self):
     def rank(self):
         return self.gen().degree()
 
+    def velu(self, candidate):
+        if not candidate in self.ore_polring():
+            raise TypeError('The candidate must be in the Ore polynomial ' \
+                    'ring')
+        # There are two main ways to give the result. The first way is
+        # to return the Drinfeld module generated by the right-quotient
+        # of `candidate * self(X)` right-divided by `candidate`. The
+        # second way is to recursively find the coefficients (see
+        # arXiv:2203.06970, Eq. 1.1). For now, the former is
+        # implemented, as it is very easy to write.
+        if candidate == 0:
+            return None
+        if not self.characteristic().degree().divides(candidate.valuation()):
+            return None
+        q, r = (candidate * self.gen()).right_quo_rem(candidate)
+        if r != 0:
+            return None
+        else:
+            return FiniteDrinfeldModule(self.polring(), q, self.characteristic())
+
     ##########################
     # Special Sage functions #
     ##########################