gh-39449: Magma padics
    
add back and forth conversion of simple padic Rings beween magma and
sage.

This is only for parents, not for elements.

The correspondance of implementations is not clear. Does magma use
relative or absolute capped precision ?

### 📝 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: #39449
Reported by: Frédéric Chapoton
Reviewer(s): Xavier Caruso

Author: Release Manager

src/sage/ext_data/magma/sage/basic.m

@@ -125,7 +125,7 @@ intrinsic Sage(X::RngIntRes) -> MonStgElt, BoolElt
   return Sprintf("Zmod(%o)", Characteristic(X)), false;
 end intrinsic;
 
-/* Approximate fields */
+/* Approximate real and complex fields */
 
 intrinsic Sage(X::FldRe) -> MonStgElt, BoolElt
 {}
@@ -147,6 +147,34 @@ intrinsic Sage(X::FldComElt) -> MonStgElt, BoolElt
   return Sprintf("%o([%o, %o])", Sage(Parent(X)), Sage(Real(X)), Sage(Imaginary(X))), true;
 end intrinsic;
 
+/* p-adic rings and fields */
+
+intrinsic Sage(X::RngPad) -> MonStgElt, BoolElt
+{p-adic rings, either free precision model or exact model}
+    prec := Precision(X);
+    if Type(prec) eq Infty then
+      return Sprintf("Zp(%o, %o, 'relaxed')", Sage(Prime(X)), Sage(prec), false;
+    else
+      return Sprintf("Zp(%o, %o, 'capped-abs')", Sage(Prime(X)), Sage(prec)), false;
+    end if;
+end intrinsic;
+
+intrinsic Sage(X::FldPad) -> MonStgElt, BoolElt
+{p-adic fields, either free precision model or exact model}
+    prec := Precision(X);
+    if Type(prec) eq Infty then
+      return Sprintf("Qp(%o, %o, 'relaxed')", Sage(Prime(X)), Sage(prec)), false;
+    else
+      return Sprintf("Qp(%o, %o, 'capped-rel')", Sage(Prime(X)), Sage(prec)), false;
+    end if;
+end intrinsic;
+
+intrinsic Sage(X::RngPadRes) -> MonStgElt, BoolElt
+{fixed precision model}
+    return Sprintf("Zp(%o, %o, 'fixed-mod')", Sage(Prime(X)), Sage(Precision(X))), false;
+end intrinsic;
+
+
 /* Polynomials */
 
 intrinsic SageNamesHelper(X::.) -> MonStgElt

src/sage/rings/padics/padic_base_leaves.py

@@ -404,6 +404,19 @@ def _convert_map_from_(self, R):
                 from sage.rings.padics.padic_generic import ResidueLiftingMap
                 return ResidueLiftingMap._create_(R, self)
 
+    def _magma_init_(self, magma):
+        """
+        Conversion to magma.
+
+        EXAMPLES::
+
+            sage: # optional - magma
+            sage: F = Qp(5,7,"capped-abs")
+            sage: magma(F)
+            5-adic field mod 5^7
+        """
+        return f"pAdicRing({self.prime()},{self.precision_cap()})"
+
 
 class pAdicRingFloatingPoint(pAdicRingBaseGeneric, pAdicFloatingPointRingGeneric):
     r"""
@@ -607,6 +620,19 @@ def _convert_map_from_(self, R):
                 from sage.rings.padics.padic_generic import ResidueLiftingMap
                 return ResidueLiftingMap._create_(R, self)
 
+    def _magma_init_(self, magma):
+        """
+        Conversion to magma.
+
+        EXAMPLES::
+
+            sage: # optional - magma
+            sage: F = Zp(5,7,"fixed-mod")
+            sage: magma(F)
+            Quotient of the 5-adic ring modulo the ideal generated by 5^7
+        """
+        return f"pAdicQuotientRing({self.prime()},{self.precision_cap()})"
+
 
 class pAdicFieldCappedRelative(pAdicFieldBaseGeneric, pAdicCappedRelativeFieldGeneric):
     r"""
@@ -719,6 +745,19 @@ def _convert_map_from_(self, R):
                 from sage.rings.padics.padic_generic import ResidueLiftingMap
                 return ResidueLiftingMap._create_(R, self)
 
+    def _magma_init_(self, magma):
+        """
+        Conversion to magma.
+
+        EXAMPLES::
+
+            sage: # optional - magma
+            sage: F = Qp(5,7,"capped-rel")
+            sage: magma(F)
+            5-adic field mod 5^7
+        """
+        return f"pAdicField({self.prime()},{self.precision_cap()})"
+
     def random_element(self, algorithm='default'):
         r"""
         Return a random element of ``self``, optionally using the ``algorithm``