gh-35035: make _multiple_x_*() methods work for all n≠0
    
Currently, these methods expect $n\geq2$ for no real reason. This patch
makes them work for all $n\neq0$.
    
URL: #35035
Reported by: Lorenz Panny
Reviewer(s): John Cremona

Author: Release Manager

src/sage/schemes/elliptic_curves/ell_generic.py

@@ -1991,6 +1991,14 @@ def _multiple_x_numerator(self, n, x=None):
 
         EXAMPLES::
 
+            sage: E = EllipticCurve([1,2])
+            sage: E._multiple_x_numerator(3)
+            x^9 - 12*x^7 - 192*x^6 + 30*x^5 - 48*x^4 + 228*x^3 + 96*x^2 + 393*x + 528
+            sage: E._multiple_x_numerator(-3)
+            x^9 - 12*x^7 - 192*x^6 + 30*x^5 - 48*x^4 + 228*x^3 + 96*x^2 + 393*x + 528
+
+        ::
+
             sage: E = EllipticCurve("37a")
             sage: P = E.gens()[0]
             sage: x = P[0]
@@ -2043,9 +2051,9 @@ def _multiple_x_numerator(self, n, x=None):
             sage: E._multiple_x_numerator(5)
             x^25 + 65037*x^23 + 55137*x^22 + ... + 813*x^2 + 10220*x + 42539
         """
-        n = rings.Integer(n)
-        if n < 2:
-            raise ValueError("n must be at least 2")
+        n = rings.Integer(n).abs()
+        if not n:
+            raise ValueError("n must be nonzero")
 
         if x is None:
             try:
@@ -2061,6 +2069,9 @@ def _multiple_x_numerator(self, n, x=None):
             cache = None
             xx = x
 
+        if n == 1:
+            return xx
+
         polys = self.division_polynomial_0([-2,-1,n-1,n,n+1], x)
 
         if n % 2 == 0:
@@ -2101,6 +2112,14 @@ def _multiple_x_denominator(self, n, x=None):
 
         EXAMPLES::
 
+            sage: E = EllipticCurve([1,2])
+            sage: E._multiple_x_denominator(3)
+            9*x^8 + 36*x^6 + 144*x^5 + 30*x^4 + 288*x^3 + 564*x^2 - 48*x + 1
+            sage: E._multiple_x_denominator(-3)
+            9*x^8 + 36*x^6 + 144*x^5 + 30*x^4 + 288*x^3 + 564*x^2 - 48*x + 1
+
+        ::
+
             sage: E = EllipticCurve("43a")
             sage: P = E.gens()[0]
             sage: x = P[0]
@@ -2128,9 +2147,9 @@ def _multiple_x_denominator(self, n, x=None):
             sage: E._multiple_x_denominator(5)
             25*x^24 + 3100*x^22 + 19000*x^21 + ... + 24111*x^2 + 52039*x + 56726
         """
-        n = rings.Integer(n)
-        if n < 2:
-            raise ValueError("n must be at least 2")
+        n = rings.Integer(n).abs()
+        if not n:
+            raise ValueError("n must be nonzero")
 
         if x is None:
             try: