axelexic · January 28, 2019 06:04 · Jan 28, 2019
diff --git a/nTorsonIsogeny b/nTorsonIsogeny
@@ -0,0 +1,52 @@
+def normalize(polyList, fx):
+    const_term  = 0
+    y_term = 0
+
+    for i, v in enumerate(polyList):
+        xterm = polyList[i]
+        if i % 2 == 0:
+            expo = int(i/2)
+            const_term = const_term + xterm * (fx ^ expo) 
+        else:
+            expo = int(int(i-1)/2)
+            y_term = y_term + xterm * (fx ^ expo)
+
+    return (y_term,const_term)
+
+
+def nTorsionIsogney(curve, c):
+    A,B          = curve.a4(), curve.a6()
+    fp           = curve.base_ring()
+    divP_Plus_2  = curve.division_polynomial(c+2,two_torsion_multiplicity=1)
+    divP_Plus_1  = curve.division_polynomial(c+1,two_torsion_multiplicity=1)
+    divP_minus_1 = curve.division_polynomial(c-1,two_torsion_multiplicity=1)
+    divP_minus_2 = curve.division_polynomial(c-2,two_torsion_multiplicity=1)
+    divP         = curve.division_polynomial(c,two_torsion_multiplicity=1)
+
+    if c % 2 == 0:
+        (X,Y)        = divP.variables()
+    else:
+        (X,Y)        = divP_Plus_1.variables()
+
+    numPhi       = (X*divP*divP - divP_Plus_1*divP_minus_1).polynomial(Y).list()
+    denPhi       = (divP*divP).polynomial(Y).list()
+    numOmega     = (divP_Plus_2*divP_minus_1*divP_minus_1 - 
+                    divP_minus_2*divP_Plus_1*divP_Plus_1).polynomial(Y).list()
+    denOmega     =  (4*Y*divP*divP*divP).polynomial(Y).list()
+
+    fx           = X^3+A*X+B;
+    (numX,numCX)  = normalize(numPhi, fx)
+    (denX,denCX)  = normalize(denPhi, fx)
+    (numY,numCY)  = normalize(numOmega, fx)
+    (denY,denCY)  = normalize(denOmega, fx)
+    return (((numCX + Y*numX),(denCX + Y*denX)), 
+            ((numCY + Y*numY),(fx*(denCY + Y*denY)/Y)))
+
+Fp = GF(43)
+FpBar=Fp.algebraic_closure();
+E  = EllipticCurve(Fp, [1,3])
+g = E.gens()[0];
+kTor = 8;
+((numX,denX), (numY, denY))=nTorsionIsogney(E, kTor)
+numX=numX.change_ring(FpBar)
+print (numX(g[0],g[1])/denX(g[0],g[1]), (g[1]*numY(g[0],g[1])/denY(g[0],g[1])))