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+from tabulate import tabulate
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+# This is a more usable version of valuate.sage, less instructional
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+
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+K.<x, y> = GF(11)[]
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+Px, Py = K(2), K(4)
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+S = K.quotient(y^2 - x^3 - 4*x).fraction_field()
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+X, Y = S(x), S(y)
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+
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+EC_A = 4
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+EC_B = 0
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+EC = y^2 - x^3 - EC_A*x - EC_B
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+
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+original_f = (y - 2*x)^2
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+b0, b1, b2 = [(x - Px), (y - Py), 1]
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+
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+# Return components for basis
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+def decomp(f):
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+ a0, r = f.quo_rem(b0)
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+ a1, r = r.quo_rem(b1)
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+ a2, r = r.quo_rem(b2)
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+ assert r == 0
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+ return [a0, a1, a2]
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+
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+def comp(comps):
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+ return sum(a*b for a, b in zip(comps, (b0, b1, b2)))
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+
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+assert comp(decomp(original_f)) == original_f
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+
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+# so we can replace (y - Py) with this
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+Ef = b0^2 + binomial(3,2)*Px*b0^1 + (3*Px^2 + EC_A)
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+Eg = (y + Py)
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+assert EC == b1*Eg - b0*Ef
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+
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+def apply_reduction(a, g):
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+ assert a[2] == 0
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+ a[0] = a[0]*Eg + a[1]*Ef
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+ a[1] = 0
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+ g[0] *= Eg
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+
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+k = 0
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+a = [original_f, 0, 0]
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+g = [1]
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+
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+table = []
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+table.append(("", "a", "g", "k"))
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+
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+def log(step_name, a, g, k):
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+ table.append((step_name, str(a), str(g), k))
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+
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+log("start", a, g, k)
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+
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+while True:
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+ f = a[0]
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+ a = decomp(f)
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+ log("decomp", a, g, k)
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+
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+ # Check remainder
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+ if a[2] != 0:
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+ break
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+
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+ # We can apply a reduction
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+ k += 1
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+
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+ apply_reduction(a, g)
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+ log("reduce", a, g, k)
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+
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+print(tabulate(table))
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+print(f"k = {k}")
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narodnik