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make _multiple_x_*() methods work for all n≠0 #35035

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Mar 3, 2023
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make _multiple_x_*() methods work for all n≠0 #35035

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31 changes: 25 additions & 6 deletions src/sage/schemes/elliptic_curves/ell_generic.py
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Original file line number Diff line number Diff line change
Expand Up @@ -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]
Expand Down Expand Up @@ -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:
Expand All @@ -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:
Expand Down Expand Up @@ -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]
Expand Down Expand Up @@ -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:
Expand Down