Initial commit

12 files changed, 39317 insertions, 0 deletions

diff --git a/l-multiples_l-torsion/.list-3-100k.swp b/l-multiples_l-torsion/.list-3-100k.swp
new file mode 100755
index 0000000..fb46e4e
--- /dev/null
+++ b/l-multiples_l-torsion/.list-3-100k.swp
diff --git a/l-multiples_l-torsion/.test4.gp.swp b/l-multiples_l-torsion/.test4.gp.swp
new file mode 100755
index 0000000..7058df4
--- /dev/null
+++ b/l-multiples_l-torsion/.test4.gp.swp
diff --git a/l-multiples_l-torsion/.test9.gp.swp b/l-multiples_l-torsion/.test9.gp.swp
new file mode 100755
index 0000000..7fe9f0f
--- /dev/null
+++ b/l-multiples_l-torsion/.test9.gp.swp
diff --git a/l-multiples_l-torsion/.test_l-multiple.sage.swp b/l-multiples_l-torsion/.test_l-multiple.sage.swp
new file mode 100755
index 0000000..41cd1eb
--- /dev/null
+++ b/l-multiples_l-torsion/.test_l-multiple.sage.swp
diff --git a/l-multiples_l-torsion/100k-3.gp b/l-multiples_l-torsion/100k-3.gp
new file mode 100755
index 0000000..451c688
--- /dev/null
+++ b/l-multiples_l-torsion/100k-3.gp
@@ -0,0 +1,9389 @@
+
+\\ Elliptic curves downloaded from the LMFDB downloaded on 10 January 2019.
+\\ Below is a list called data. Each entry has the form:
+\\   [a1,a2,a3,a4,a6] (Weierstrass Coefficients)
+
+
+data = [\
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diff --git a/l-multiples_l-torsion/400k-3.gp b/l-multiples_l-torsion/400k-3.gp
new file mode 100755
index 0000000..43eba43
--- /dev/null
+++ b/l-multiples_l-torsion/400k-3.gp
@@ -0,0 +1,27648 @@
+
+\\ Elliptic curves downloaded from the LMFDB downloaded on 11 January 2019.
+\\ Below is a list called data. Each entry has the form:
+\\   [a1,a2,a3,a4,a6] (Weierstrass Coefficients)
+
+
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diff --git a/l-multiples_l-torsion/bunch_of_curves.gp b/l-multiples_l-torsion/bunch_of_curves.gp
new file mode 100755
index 0000000..e100c56
--- /dev/null
+++ b/l-multiples_l-torsion/bunch_of_curves.gp
@@ -0,0 +1,2022 @@
+
+\\ Elliptic curves downloaded from the LMFDB downloaded on 31 January 2019.
+\\ Below is a list called data. Each entry has the form:
+\\   [a1,a2,a3,a4,a6] (Weierstrass Coefficients)
+
+\\~2000 curves with rank 1 and conductor <1000
+
+data = [\
+[0,0,1,-1,0],\
+[0,1,1,0,0],\
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+[1,1,0,-428,2832],\
+[1,1,0,-108,-432],\
+[1,1,0,652,16008],\
+[1,1,0,-47,99],\
+[1,1,0,3,9],\
+[1,1,0,-37442,2585844],\
+[1,1,0,2238,181236],\
+[1,0,1,-2013,-12344],\
+[1,0,1,467,-1432],\
+[1,1,1,-161,119],\
+[1,1,1,39,39],\
+[0,-1,1,-3,-1],\
+[0,1,1,-399,-3184],\
+[1,0,1,-3,0],\
+[0,1,1,-1,-4],\
+[0,0,0,-7491,249550],\
+[0,0,0,-471,3850],\
+[0,0,0,-66,-119],\
+[0,0,0,69,12166],\
+[0,0,0,-51,94],\
+[0,0,0,9,10],\
+[0,0,0,-30,29],\
+[0,0,0,105,218],\
+[0,0,0,-615,5866],\
+[0,0,0,-30,133],\
+[1,0,1,-11085,446696],\
+[1,0,1,-365,13608],\
+[1,0,1,-4,-2],\
+[1,1,1,-56,-135],\
+[0,1,1,4,14],\
+[0,-1,1,-321,-9817],\
+[1,0,1,-6,-5],\
+[1,0,1,9,-23],\
+[0,-1,0,-20,40],\
+[0,1,0,-31516,-1956716],\
+[0,1,0,-7076,226340],\
+[1,0,0,-65,201],\
+[1,0,0,146,37508],\
+[0,1,0,-64,180],\
+[0,1,0,8,-12],\
+[0,-1,0,-9,-8],\
+[0,-1,0,31,-68],\
+[0,1,0,8,-44],\
+[0,1,0,-276,1676],\
+[0,1,0,4,-4],\
+[0,1,0,-400,-3308],\
+[0,1,0,1840,162452],\
+[1,0,1,-261,-1680],\
+[1,0,1,14,-8],\
+[1,0,1,-1751,-31352],\
+[1,0,1,-1,148],\
+[1,1,1,-2138,-306969],\
+[1,1,1,-388,2781],\
+[1,1,1,237,11281],\
+[1,-1,0,-81,783],\
+[1,-1,0,9,-27],\
+[1,-1,0,-221427,40159989],\
+[1,-1,0,-2547,63477],\
+[1,-1,0,18,202],\
+[1,-1,0,-96,-640],\
+[1,-1,1,1273,-3585],\
+[1,-1,1,-11,27],\
+[1,-1,1,-248,1563],\
+[0,-1,0,-641,6465],\
+[0,-1,0,-161,-639],\
+[0,-1,0,-41,105],\
+[0,-1,0,4,6],\
+[0,-1,0,-12801,-553215],\
+[0,-1,0,-801,-8415],\
+[0,-1,0,-321,-18879],\
+[0,-1,0,-81,81],\
+[0,-1,0,-61,205],\
+[0,-1,0,319,321],\
+[0,-1,0,-225,-1215],\
+[0,-1,0,-25,25],\
+[0,-1,0,-20,42],\
+[0,-1,0,95,97],\
+[0,1,0,-1921,31775],\
+[0,1,0,-321,-1665],\
+[0,1,0,-121,455],\
+[0,1,0,4,30],\
+[0,1,0,-12801,553215],\
+[0,1,0,-801,8415],\
+[0,1,0,-321,18879],\
+[0,1,0,-81,-81],\
+[0,1,0,-61,-205],\
+[0,1,0,319,-321],\
+[0,1,0,-1985,-19617],\
+[0,1,0,-905,9975],\
+[0,1,0,-900,10098],\
+[0,1,0,95,31775],\
+[1,1,0,-72,-90],\
+[1,1,0,18,0],\
+[1,1,0,-3346,63700],\
+[1,1,0,334,5556],\
+[1,1,0,-250264,48082240],\
+[1,1,0,-14744,836928],\
+[1,0,1,-361,2564],\
+[1,0,1,-1,116],\
+[1,1,1,-79134,-8601153],\
+[1,1,1,-17714,900047],\
+[1,1,1,-5074,-128689],\
+[1,1,1,-1154,12431],\
+[1,1,1,126,1167],\
+[1,1,1,6266,-609505],\
+[0,0,0,-484,-5324],\
+[0,0,0,-11,22],\
+[0,1,0,15,-13],\
+[0,0,0,0,-972],\
+[0,0,0,0,36],\
+[0,0,0,0,-243],\
+[0,0,0,0,9],\
+[0,1,1,-11373,-470630],\
+[0,1,1,-253,441],\
+[0,1,1,-203,1048],\
+[1,-1,1,51,117],\
+[1,-1,1,3,-3],\
+[1,1,1,-91,297],\
+[1,1,1,-5,3],\
+[0,-1,1,-8,-82],\
+[0,1,1,-4758,128144],\
+[1,0,0,12,117],\
+[0,-1,1,-83,3818],\
+[0,1,1,-3,29],\
+[1,1,0,-3250000,-2256492875],\
+[1,1,0,-203125,-35321000],\
+[1,1,0,-198250,-37090625],\
+[1,1,0,-13000,-528125],\
+[1,1,0,-2875,49000],\
+[1,1,0,-2750,54375],\
+[1,1,0,5250,284625],\
+[1,1,0,15125,-2468750],\
+[1,0,1,-46,-127],\
+[0,0,0,1,-6],\
+[1,0,1,-5,2],\
+[1,1,1,-121,455],\
+[1,0,0,-132,144],\
+[1,1,0,-14646,-688345],\
+[1,1,0,-14041,-747030],\
+[0,-1,0,-39461,3030385],\
+[0,-1,0,-261,8065],\
+[0,1,0,19,-1],\
+[1,-1,1,-74,262],\
+[1,-1,0,36,81],\
+[1,0,1,-22,40],\
+[0,-1,0,-369,4293],\
+[0,1,0,7,27],\
+[0,1,1,-20,24],\
+[1,1,0,-10407,-413003],\
+[1,1,0,-276,1616],\
+[1,1,0,44,5520],\
+[1,0,0,3467,-83679],\
+[1,-1,1,-1,17],\
+[1,0,0,8,16],\
+[1,0,0,-6909,79524],\
+[1,0,0,1596,9783],\
+[0,0,0,16,36],\
+[0,0,0,-362249,165197113],\
+[1,-1,0,-15,25],\
+[1,-1,0,15,91],\
+[1,-1,0,-34965,2525175],\
+[1,-1,0,-10575,-415935],\
+[1,-1,0,-2295,35721],\
+[1,-1,0,-675,-6075],\
+[1,-1,0,45,-459],\
+[1,-1,0,4455,201771],\
+[1,-1,1,-14423,-663069],\
+[1,-1,1,-923,-9669],\
+[1,-1,1,-203,987],\
+[1,-1,1,1057,-46893],\
+[1,-1,1,-96608,-11533373],\
+[1,-1,1,-95528,-11804669],\
+[1,-1,1,-1568,-4669],\
+[1,-1,1,6112,-41533],\
+[1,0,1,-1,4],\
+[1,0,1,-1611,-39690],\
+[1,0,1,164,922],\
+[1,0,0,-52238,-4587140],\
+[1,0,0,-52198,-4594524],\
+[1,0,0,-3238,65508],\
+[1,0,0,-678,-5660],\
+[1,-1,1,-1516,23091],\
+[1,-1,1,-96,371],\
+[1,-1,1,-16,-13],\
+[1,-1,1,44,1267],\
+[0,1,1,-15,19],\
+[0,1,1,85,64],\
+[0,1,0,-12,36],\
+[0,1,0,108,-876],\
+[0,1,0,164,-8764],\
+[0,-1,1,-18,36],\
+[1,-1,1,-8,10],\
+[1,-1,0,-69,-208]]
diff --git a/l-multiples_l-torsion/l-multiples_l-torsion.pdf b/l-multiples_l-torsion/l-multiples_l-torsion.pdf
new file mode 100755
index 0000000..c13fc12
--- /dev/null
+++ b/l-multiples_l-torsion/l-multiples_l-torsion.pdf
diff --git a/l-multiples_l-torsion/l-multiples_l-torsion.tex b/l-multiples_l-torsion/l-multiples_l-torsion.tex
new file mode 100755
index 0000000..a07078b
--- /dev/null
+++ b/l-multiples_l-torsion/l-multiples_l-torsion.tex
@@ -0,0 +1,148 @@
+\documentclass[10pt,a4paper]{article}
+\usepackage[utf8]{inputenc}
+\usepackage{amsmath}
+\usepackage{amsthm}
+\usepackage[all]{xy}
+\usepackage{amsfonts}
+\usepackage{color}
+\usepackage{amssymb}
+\usepackage{float}
+\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}
+\DeclareMathOperator{\Tr}{Tr}
+\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}
+\DeclareMathOperator{\mat}{Mat}
+\DeclareMathOperator{\ab}{ab}
+
+\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}
+\newtheorem{proposition}[lemma]{Proposition}
+\newtheorem{corollary}[lemma]{Corollary}
+\theoremstyle{definition}
+\newtheorem{remark}[lemma]{Remark}
+
+\title{Rational $\ell$-multiples of points over the $\ell$-torsion field}
+\author{Sebastiano Tronto}
+
+\begin{document}
+
+\maketitle
+
+Let $\ell$ be a rational prime and let $A$ be an abelian variety of dimension $d$ over a number field $K$. Let $K_\ell=K(A[\ell])$ and let $\mathcal{T}_1=\gal(K_\ell\,|\,K)$.
+
+\begin{remark}
+For $m\geq 1$ we have $\#\gl_m(\F_\ell)=\prod_{i=0}^{\ell-1}(\ell^m-\ell^i)$. In fact, elements of $\gl_m(\F_\ell)$ are in bijection with bases of $\F^m_\ell$, and counting basis of a vector space over a finite field is a simple combinatorics exercise: first we pick any non-zero vector ($\ell^n-1$ possibilities), then we pick any vector that is not in the $\F_\ell$-span on the first one ($\ell^n-\ell$ possibilities), then a third one that is not in the span of the first two...
+
+In particular, $v_\ell\left(\#\gl_m(\F_\ell)\right)=\frac{1}{2}m(m-1)$.
+\end{remark}
+
+Compare the next lemma with \cite{jr}, Lemma 3.7.
+
+\begin{lemma}
+There is an exact sequence
+\begin{align*}
+0\to \ell A(K) \to A(K)\cap \ell A(K_{\ell})\to H^1(\mathcal{T}_1,A[\ell]).
+\end{align*}
+In particular, if $H^1(\mathcal{T}_1,A[\ell])=0$ we have $A(K)\cap \ell A(K_\ell)=\ell A(K)$.
+\begin{proof}
+Consider the short exact sequence of $\mathcal{T}_1$-modules
+\begin{align*}
+0\to A[\ell](K_{\ell})\to A(K_{\ell})\to \ell A(K_{\ell})\to 0
+\end{align*}
+and the induced long exact sequence in cohomology (i.e. take $H^*(\mathcal{T}_1,-)$)
+\begin{align*}
+0\to A[\ell](K)\to A(K)\to A(K)\cap\ell A(K_{\ell})\to H^1(\mathcal{T}_1,A[\ell])\to\cdots
+\end{align*}
+and the thesis follows by noticing that $A(K)/A[\ell](K)\cong \ell A(K)$.
+\end{proof}
+\end{lemma}
+
+This leads us to study the group $H^1(\mathcal{T}_1,A[\ell])$. In particular, we would like to know in which cases it is trivial (and so we are happy). In particular, the case of elliptic curves has been completely solved if $K=\kiu$, and there are rather complete results if $K\cap \kiu(\zeta_\ell)=\kiu$ (see \cite{lawson}).
+
+It is in fact possible that some point $\alpha\in A(K)$ is not $\ell$-divisible in $A(K)$, but becomes $\ell$-divisible in $A(K_\ell)$. The smallest example I have found is the point 
+$(23769/400, 3529853/8000)$ on the elliptic curve over $\kiu$ with Cremona Label $17739g1$. A gp script that finds all the 12 examples of elliptic curves with conductor $<10^5$ with a generator of the free part of the group of rational points that becomes $3$-divisible over $K_3$ can be found in \texttt{test3.gp}.
+\\
+
+The question now becomes: ``How much'' can a point $\alpha\in A(K)$ become $\ell$-divisible in $A(K_{\ell^\infty})$? That is, can we find an (explicit) $N$ such that there is no $\beta\in A(K_{\ell^\infty})$ with $\alpha=\ell^n\beta$ for $n\geq N$?
+
+%\textbf{We restrict to the case where $A=E$ is an elliptic curve without complex multiplication}. Following the proof of Theorem 5.2 of \cite{jr} (using Lemma 3.6 of the same article), we see that $A(K)\cap \ell A(K_{\ell^n})=A(K)\cap \ell A(K_{\ell^{n-1}})$ for any $n\geq n_0$, where $n_0$ is such that $A$ satisfies maximal growth of the torsion part starting from $n_0$. So we can replace $K_{\ell^{\infty}}$ with the finite extension $K_{\ell^{n_0}}$.
+
+%Second question: work in progress! (Latest idea: use heights and the fact that that $P$ does not become ``more divisible'' by $\ell$ after a certain point; see descent part in the proof of Mordell-Weil).
+
+(If we could say $A(K)\cap\ell A(K_{\ell^\infty})=A(K)\cap\ell A(K_{\ell^{n_0}})$ for some $n_0$, then we could use Petsche's results to explicitly bound $N$ in terms of the height of $P$)
+
+\begin{thebibliography}{[99]}
+\bibitem{jr} R. Jones, J. Rouse, \emph{Galois Theory of Iterated Endomorphisms}, preprint(?).
+%\bibitem{lom} D. Lombardo, A. Perucca, \emph{Reductions of Points on Algebraic Groups}, preprint.
+%\bibitem{milne-ft} J. S. Milne, \emph{Fields and Galois Theory}, Online notes.
+\bibitem{lawson} T. Lawson, C. Wuthrich, \emph{Vanishing of some Galois cohomology groups for elliptic curves}, preprint(?).
+\end{thebibliography} 
+
+\end{document}
\ No newline at end of file
diff --git a/l-multiples_l-torsion/test3-extended.gp b/l-multiples_l-torsion/test3-extended.gp
new file mode 100755
index 0000000..5a99a7a
--- /dev/null
+++ b/l-multiples_l-torsion/test3-extended.gp
@@ -0,0 +1,40 @@
+read("400k-3.gp");
+/* Testing all ECs with a 3-torsion point, conductor < 4*10^5 and rank > 0 */
+/* Taken from LMFBD */
+
+S = List();
+
+{
+for( i = 1, length( data ), 
+  if( i % 100 == 1, print( "Tested ", i-1, " curves." ); );
+  coeffs = data[i];
+  E = ellinit( coeffs );
+  f3 = elldivpol( E, 3 ); /* 3-division polynomial */
+  eqn = y^2 + E[1]*x*y + E[3]*y - x^3 - E[2]*x^2 - E[4]*x - E[5]; /* eqn of E */
+  res = substpol( polresultant( f3, eqn ), y, x ); /* Keep x for consistency */
+  K3 = nfinit( polredbest( nfsplitting( res, 6 ) ) );
+  E_ext = ellinit( coeffs, K3 );
+  Egens = ellgenerators(E);
+  for( j = 1, length( Egens ), 
+    P = Egens[j];
+    if( ellisdivisible( substpol(E_ext,x,y), P, 3, &Q ),
+      cremona_label = ellidentify(E)[1][1];
+      listput( S, [cremona_label, P, Q] );
+      write( "pari-output-ext", cremona_label );
+      write( "pari-output-ext", P );
+      write( "pari-output-ext", Q );
+      print( "Found EC ", cremona_label, " with point ", P );
+      /* Check for 3-divisibility up to torsion */
+      T = elltors(E)[3][1];
+      if( ellisdivisible( E, elladd(E,P,T), 3 ),
+        print( "P+T_1 is 3-divisible!" );
+      );
+      T = ellmul(E,T,2);    
+      if( ellisdivisible( E, elladd(E,P,T), 3 ),
+        print( "P+T_2 is 3-divisible!" );
+      );  
+    );
+  );
+);
+}
+
diff --git a/l-multiples_l-torsion/test4.gp b/l-multiples_l-torsion/test4.gp
new file mode 100755
index 0000000..4ef8f02
--- /dev/null
+++ b/l-multiples_l-torsion/test4.gp
@@ -0,0 +1,36 @@
+read("bunch_of_curves.gp");
+
+{
+for( i = 1, length( data ), 
+  print( "Testing curve with coefficients ", data[i] );
+  E = ellinit( data[i] );
+  f4 = elldivpol( E, 4 ); /* 4-division polynomial */
+  eqn = y^2 + E[1]*x*y + E[3]*y - x^3 - E[2]*x^2 - E[4]*x - E[5]; /* eqn of E */
+  res = substpol( polresultant( f4, eqn ), y, x ); /* Keep x for consistency */
+  print( "Computing K4..." );
+  K4 = nfinit( nfsplitting( res, 96 ) );
+  print( "Computed!" );
+  E_ext = ellinit( E[1..5], K4 );
+  Egens = ellgenerators(E);
+  for( j = 1, length( Egens ), 
+    P = Egens[j];
+    if( ellisdivisible( substpol(E_ext,x,y), P, 2, &Q ),
+      print( "Found EC ", cremona_label, " with point ", P, 
+             " and 2-division point ", Q, " over K4" );
+      if( ellisdivisible( substpol(E_ext,x,y), P, 4, &Q ),
+        print( "Even 4-divisible! 4-division point: ", Q );
+      );
+      /* Check for 3-divisibility up to torsion */
+      /* T = elltors(E)[3][1];
+      if( ellisdivisible( E, elladd(E,P,T), 3 ),
+        print( "P+T_1 is 3-divisible!" );
+      );
+      T = ellmul(E,T,2);    
+      if( ellisdivisible( E, elladd(E,P,T), 3 ),
+        print( "P+T_2 is 3-divisible!" );
+      ); */ 
+    );
+  );
+);
+}
+
diff --git a/l-multiples_l-torsion/test9.gp b/l-multiples_l-torsion/test9.gp
new file mode 100755
index 0000000..47b00ab
--- /dev/null
+++ b/l-multiples_l-torsion/test9.gp
@@ -0,0 +1,34 @@
+/* Candidate counterex mod3 not surj, unexpected density, rank >=1 */
+label = [ "220a3", "220a4", "660c3", "660c4", "770f3", "770f4" ];
+
+{
+for( i = 1, length( data ), 
+  print( "Testing curve ", i, ", label ", label[i] );
+  E = ellinit( label[i] );
+  f9 = elldivpol( E, 9 ); /* 3-division polynomial */
+  eqn = y^2 + E[1]*x*y + E[3]*y - x^3 - E[2]*x^2 - E[4]*x - E[5]; /* eqn of E */
+  res = substpol( polresultant( f9, eqn ), y, x ); /* Keep x for consistency */
+  print( "Computing K9..." );
+  K9 = nfinit( nfsplitting( res, 3^5*2 ) ); /* mo3 is at most 6 */
+  print( "Computed!" );
+  E_ext = ellinit( E[1..5], K9 );
+  Egens = ellgenerators(E);
+  for( j = 1, length( Egens ), 
+    P = Egens[j];
+    if( ellisdivisible( substpol(E_ext,x,y), P, 3, &Q ),
+      print( "Found EC ", cremona_label, " with point ", P, 
+             " and 3-division point ", Q, " over K9" );
+      /* Check for 3-divisibility up to torsion */
+      /* T = elltors(E)[3][1];
+      if( ellisdivisible( E, elladd(E,P,T), 3 ),
+        print( "P+T_1 is 3-divisible!" );
+      );
+      T = ellmul(E,T,2);    
+      if( ellisdivisible( E, elladd(E,P,T), 3 ),
+        print( "P+T_2 is 3-divisible!" );
+      ); */ 
+    );
+  );
+);
+}
+