From 5ea79c7ae0d44686f1df05c4a016652afbe58968 Mon Sep 17 00:00:00 2001
From: Sebastiano Tronto <sebastiano@tronto.net>
Date: Sun, 14 Jun 2026 09:58:21 +0200
Subject: Initial commit

---
 misc_code/2_division.sage | 126 ++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 126 insertions(+)
 create mode 100755 misc_code/2_division.sage

(limited to 'misc_code/2_division.sage')

diff --git a/misc_code/2_division.sage b/misc_code/2_division.sage
new file mode 100755
index 0000000..68cea80
--- /dev/null
+++ b/misc_code/2_division.sage
@@ -0,0 +1,126 @@
+
+R1.<x> = PolynomialRing(QQ)
+R2.<x,y> = PolynomialRing(QQ)
+
+def extended_field( f, A, B, deg_mult ):
+    # f: a polynomial whose roots are the x-coordinates of some points of an
+    #    elliptic curve E: y^2 = x^3 + Ax + B.
+    # return value: a field containing the x and y coordinates of those points
+    # deg_mult: a positive integer known to be a multiple of the degree of the
+    #    extended field.
+    #
+    # This function uses the properties of resultants (I can provide a pdf
+    # explaining how it works).
+    # 
+    # It is useful to compute, e.g., the fields obtained by adjoining the
+    # coordinates of the n-division points of a point (using the n-uplication
+    # formulas to get the required polynomials).
+    #
+    # When used to compute the 2-division fields, it gives the same output as
+    # E.division_field(2).
+
+    
+    g = y^2 - x^3 - A*x - B
+    res = f.resultant(g,x)
+    res = res.subs(y=x)
+
+    #K.<b> = f.splitting_field()
+    #print aux
+    print "+++ Computing splitting field of the following: +++"
+    
+    pol = R1(res*f)
+    print pol
+    
+    K.<b> = NumberField( R( pari(pol).nfsplitting(deg_mult) ) )
+
+    return K
+
+L = []
+
+for A in range(1,9):
+    for B in range(1,9):
+
+        print "Current list of examples:", len(L), "elements. List:"
+        print L
+        
+        E = EllipticCurve([0,0,0,A,B])
+        print "*************************"
+        print E
+        print "of rank ", E.rank(), "with (some) points of infinite order:", E.gens()
+        print "CM:", E.has_cm()
+
+        rep = E.galois_representation()
+        print "mod 2 rep is surjective:", rep.is_surjective(2)
+
+        if E.rank() == 0:
+            print "Stopping because rank 0"
+            print ""
+            continue
+        if len(E.gens()) == 0:
+            print "Stopping because no points of infinite order found"
+            print ""
+            continue
+        if E.has_cm():
+            print "Stopping because CM"
+            print ""
+            print ""
+            continue
+        if not rep.is_surjective(2):
+            print "Stopping because mod 2 rep is not surjective"
+            print ""
+            continue
+    
+        K_2.<a> = E.division_field(2)
+        print "2-division field:", K_2
+        P = E(0)
+        flag = False
+        for P in E.gens():
+            if len(P.division_points(2)) == 0:
+                flag = True
+                break
+        if not flag:
+            print "Stopping because the points found are 2-divisible"
+            print ""
+            continue
+        print "Taking the 2-division of P =", P
+
+        # The following polynomial is derived from the duplication formula
+        # (Silverman, p.54) using the x-coordinate of the 2-division point as
+        # an indeterminate.x
+        f_P = R1(x^4 - 4*P[0]*x^3 - 2*A*x^2 - 4*(2*B + A*P[0])*x + A^2 - 4*B*P[0])
+        #print f_P.roots()
+
+        M = extended_field( f_P, A, B, 24 ) # The 2-division field of P
+        print "2-division field of P:", M
+
+        if M.degree() != 24:
+            print "Stopping because 2-division of P is too small"
+            print ""
+            continue
+
+        div_pol_4 = E.division_polynomial(4)
+
+        #if div_pol_4.splitting_field('zz').degree() < 48:
+        #    print "Stopping because splitting field of div_pol_4 < 48"
+        #    exit()
+
+        K_4 = extended_field( div_pol_4, A, B, 96 )
+        print "4-division field:", K_4
+
+        if K_4.degree() != 96:
+            print "Stopping because mod 4 representation not surjective"
+            print ""
+            continue
+
+        if len(f_P.roots(ring=K_4)) == 0:
+            print "Stopping because M is not contained in K_4"
+            print ""
+            continue
+
+        print "----------------"
+        print "|Example Found!|"
+        print "----------------"
+        L.append((A,B))
+
+        print "*************************"
+        print ""
-- 
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