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diff --git a/necessity_52b/2_division.sage b/necessity_52b/2_division.sage
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+
+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 ""
diff --git a/necessity_52b/2_division_counterex.sage b/necessity_52b/2_division_counterex.sage
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+++ b/necessity_52b/2_division_counterex.sage
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+
+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()
diff --git a/necessity_52b/necessity_52b.pdf b/necessity_52b/necessity_52b.pdf
new file mode 100755
index 0000000..1e2d18e
--- /dev/null
+++ b/necessity_52b/necessity_52b.pdf
diff --git a/necessity_52b/necessity_52b.tex b/necessity_52b/necessity_52b.tex
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+\documentclass[10pt,a4paper]{article}
+\usepackage[utf8]{inputenc}
+\usepackage{amsmath}
+\usepackage{amsthm}
+\usepackage{amsfonts}
+\usepackage{amssymb}
+\usepackage[a4paper, top=3cm, bottom=3cm, left=2.5cm, right=2.5cm]{geometry}
+
+\DeclareMathOperator{\alg}{alg}
+\DeclareMathOperator{\obj}{Obj}
+\DeclareMathOperator{\Hom}{Hom}
+\DeclareMathOperator{\End}{End}
+\DeclareMathOperator{\hol}{Hol}
+\DeclareMathOperator{\aut}{Aut}
+\DeclareMathOperator{\gal}{Gal}
+\DeclareMathOperator{\id}{id}
+\DeclareMathOperator{\res}{res}
+\DeclareMathOperator{\im}{Im}
+\DeclareMathOperator{\Id}{Id}
+\DeclareMathOperator{\fib}{Fib}
+\DeclareMathOperator{\spec}{Spec}
+\DeclareMathOperator{\proj}{Proj}
+\DeclareMathOperator{\trdeg}{trdeg}
+\DeclareMathOperator{\car}{char}
+\DeclareMathOperator{\Frac}{Frac}
+\DeclareMathOperator{\reduced}{red}
+\DeclareMathOperator{\real}{Re}
+\DeclareMathOperator{\imag}{Im}
+\DeclareMathOperator{\vol}{vol}
+\DeclareMathOperator{\den}{den}
+\DeclareMathOperator{\rank}{rank}
+\DeclareMathOperator{\lcm}{lcm}
+\DeclareMathOperator{\rad}{rad}
+\DeclareMathOperator{\ord}{ord}
+\DeclareMathOperator{\Br}{Br}
+\DeclareMathOperator{\inv}{inv}
+\DeclareMathOperator{\Nm}{Nm}
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+\DeclareMathOperator{\an}{an}
+\DeclareMathOperator{\op}{op}
+\DeclareMathOperator{\sep}{sep}
+\DeclareMathOperator{\unr}{unr}
+\DeclareMathOperator{\et}{\acute et}
+\DeclareMathOperator{\ev}{ev}
+\DeclareMathOperator{\gl}{GL}
+\DeclareMathOperator{\SL}{SL}
+
+\newcommand{\grp}{\textsc{Grp}}
+\newcommand{\set}{\textsc{Set}}
+\newcommand{\x}{\mathbf{x}}
+\newcommand{\naturalto}{\overset{.}{\to}}
+\newcommand{\qbar}{\overline{\mathbb{Q}}}
+\newcommand{\zbar}{\overline{\mathbb{Z}}}
+
+\newcommand{\pro}{\mathbb{P}}
+\newcommand{\aff}{\mathbb{A}}
+\newcommand{\quat}{\mathbb{H}}
+\newcommand{\rea}{\mathbb{R}}
+\newcommand{\kiu}{\mathbb{Q}}
+\newcommand{\F}{\mathbb{F}}
+\newcommand{\zee}{\mathbb{Z}}
+\newcommand{\ow}{\mathcal{O}}
+\newcommand{\mcx}{\mathcal{X}}
+\newcommand{\mcy}{\mathcal{Y}}
+\newcommand{\mcs}{\mathcal{S}}
+\newcommand{\mca}{\mathcal{A}}
+\newcommand{\mcb}{\mathcal{B}}
+\newcommand{\mcf}{\mathcal{F}}
+\newcommand{\mcg}{\mathcal{G}}
+\newcommand{\mct}{\mathcal{T}}
+\newcommand{\mcq}{\mathcal{Q}}
+\newcommand{\mcr}{\mathcal{R}}
+\newcommand{\adl}{\mathbf{A}}
+\newcommand{\mbk}{\mathbf{k}}
+\newcommand{\m}{\mathfrak{m}}
+\newcommand{\p}{\mathfrak{p}}
+
+\newcommand{\kbar}{\overline{K}}
+\newtheorem{lemma}{Lemma}
+
+\author{Sebastiano Tronto}
+\title{On the Second Condition of Theorem 5.2 ($\ell=2$)}
+
+\begin{document}
+
+\maketitle
+
+\textbf{Question:} is Lemma 1 of Lang's \emph{Elliptic Curves Diophantine Analysis}, Chapter 5 Section 5, page 117 (References/Kummer theory/LANG-book-EC.pdf) consistent with Jones-Rouse's Theorem 5.2, i.e. can we deduce that $F(\beta_1)$ is partially contained in $F(A[2])$ already?
+
+I believe the answer is no. Lang is working over a base field that contains all the $\ell^\infty$ torsion of $A$, so it doesn't apply in our case (J\&R assume surjectivity of the torsion part).
+
+A counterexample is actually given by J\&R right after the proof of the theorem (``Remark''). They don't say explicitly that $\kiu(\beta_1)\cap \kiu(A[2])=\kiu$ in this case, but I have tested this with sage (see the file \texttt{2-division-counterex.sage}): $\kiu(\beta_1)$ has degree $24$ over $\kiu$ and $\kiu(A[2])$ has degree $2$, which together imply that $[\kiu(A[2],\beta_1):\kiu(A[2])]\geq 4$, so it is maximal.
+
+The file \texttt{2-division.sage} contains some code that looks for other counterexamples (varying the parameters of a short Weierstrass equation), but it is quite slow (about 3 minutes on my pc for each elliptic curve of which it computes the 4-torsion field).
+
+\end{document}
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