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

---
 necessity_52b/2_division_counterex.sage | 59 +++++++++++++++++++++++++++++++++
 1 file changed, 59 insertions(+)
 create mode 100755 necessity_52b/2_division_counterex.sage

(limited to 'necessity_52b/2_division_counterex.sage')

diff --git a/necessity_52b/2_division_counterex.sage b/necessity_52b/2_division_counterex.sage
new file mode 100755
index 0000000..f1193fb
--- /dev/null
+++ b/necessity_52b/2_division_counterex.sage
@@ -0,0 +1,59 @@
+
+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)
+
+    #print "Splitting field of division pol:"
+    #print R1(f).splitting_field('r')
+    K.<b> = (R1(res*f)).splitting_field(degree_multiple=deg_mult)
+
+    return K
+
+
+A = -3
+B = 17/4
+
+E = EllipticCurve([0,0,0,A,B])
+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)
+
+K_2.<a> = E.division_field(2)
+print "2-division field:", K_2
+P = [4,15/2]
+
+# 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])
+
+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"
+    exit()
-- 
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