gh-39629: add magma as optional algorithm for Coxeter polynomial of posets
    
as this is a useful alternative for those who can afford to pay for
magma

### 📝 Checklist

- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
- [ ] I have linked a relevant issue or discussion.
- [x] I have created tests covering the changes.
- [x] I have updated the documentation and checked the documentation
preview.
    
URL: #39629
Reported by: Frédéric Chapoton
Reviewer(s): Travis Scrimshaw

Author: Release Manager

src/sage/combinat/posets/posets.py

@@ -4329,10 +4329,15 @@ def coxeter_transformation(self):
         """
         return self._hasse_diagram.coxeter_transformation()
 
-    def coxeter_polynomial(self):
+    def coxeter_polynomial(self, algorithm="sage"):
         """
         Return the Coxeter polynomial of the poset.
 
+        INPUT:
+
+        - ``algorithm`` -- optional (default: ``"sage"``) ;
+          the unique other option is ``"magma"``
+
         OUTPUT: a polynomial in one variable
 
         The output is the characteristic polynomial of the Coxeter
@@ -4349,11 +4354,22 @@ def coxeter_polynomial(self):
             sage: p.coxeter_polynomial()                                                # needs sage.groups sage.libs.flint
             x^6 + x^5 - x^3 + x + 1
 
+        TESTS::
+
+            sage: P = posets.PentagonPoset()
+            sage: P.coxeter_polynomial("magma")  # optional - magma
+            x^5 + x^4 + x + 1
+
         .. SEEALSO::
 
             :meth:`coxeter_transformation`, :meth:`coxeter_smith_form`
         """
-        return self._hasse_diagram.coxeter_transformation().charpoly()
+        cox_matrix = self._hasse_diagram.coxeter_transformation()
+        if algorithm == "magma":
+            from sage.interfaces.magma import magma
+            dense_matrix = magma(cox_matrix).Matrix()
+            return dense_matrix.CharacteristicPolynomial().sage()
+        return cox_matrix.charpoly()
 
     def coxeter_smith_form(self, algorithm='singular'):
         """