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\vskip 1cm{\LARGE\bf 
 A Search for High Rank Congruent Number \\
 \vskip .1in
 Elliptic Curves} \\
{
\vskip 1cm
\large
Andrej Dujella \\
Department of Mathematics\\
University of Zagreb\\
Bijeni\v{c}ka cesta 30\\
10000 Zagreb \\
Croatia\\
\href{mailto:duje@math.hr}{\tt duje@math.hr}\\
\ \\
Ali S. Janfada\footnote{The second author was
partially financed by a grant No. 009/4/87 from Urmia University.}
and Sajad Salami \\
Department of Mathematics \\
University of Urmia\\
P.O. Box 165 \\
Urmia \\
Iran \\
\href{mailto:a.sjanfada@urmia.ac.ir}{\tt a.sjanfada@urmia.ac.ir} \\
\href{mailto:asjanfada@gmail.com}{\tt asjanfada@gmail.com} \\
\href{mailto: salami.sajad@gmail.com}{\tt  salami.sajad@gmail.com}\\
}
\end{center}
\vskip .2in

\begin{abstract}
In this article, we describe a method for finding congruent number
elliptic curves with high ranks. The method involves an algorithm
based on the Monsky's formula for computing $2$-Selmer rank of
congruent number elliptic curves, and Mestre-Nagao's sum which is
used in sieving curves with potentially large ranks. We apply
this method for positive squarefree integers in two families of
congruent numbers and find some new congruent number elliptic
curves with rank $6$.
\end{abstract}

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\section{Introduction}

One of the major topics connected with elliptic curves is
construction of elliptic curves with high ranks. Several authors
considered this problem for elliptic curves with prescribed
properties and relatively high ranks. For instance, we cite
\cite{duje,kulst} for the curves with given torsion groups,
\cite{ACP2,elkis2} for the curves  $y^2=x^3+ dx$,
\cite{elkis3,rgs2} for the curves $x^3+y^3=k$ related to the
so-called taxicab problem, \cite{duje2} for the curves
$y^2=(ax+1)(bx+1)(cx+1)(dx+1)$ induced by Diophantine quadruples
$\{a,b,c,d\}$, etc. Dujella \cite{duje} collected a list of known
high rank elliptic curves with prescribed torsion groups. The
largest known rank of elliptic curves, found by N. D. Elkies in
2006, is $28$.

In this work we deal with a family of elliptic curves which are
closely  related to the classical Congruent Number problem. A
positive squarefree integer $n$ is  called a {\it congruent
number} if it is the area of a right triangle with rational sides
\seqnum{A006991}, \seqnum{A003273}. The problem of determining
congruent numbers is closely related to the curves $E_n:
y^2=x^3-n^2x$, which are called {\it congruent number elliptic
curves} or {\it CN-elliptic curves}. In fact, the positive
squarefree integer $n$ is a congruent number if and only if the
Mordell-Weil rank $r(n)$ of $E_n$ is a positive integer
 \cite[Chap. 1, Prop. 18]{kob1}.
In this case, we refer to $n$ itself as a CN-elliptic curve, which
corresponds to $E_n$. In 1972, Alter, Curtz, and Kubota \cite{Alter}
conjectured that $n \equiv 5,6,7 \pmod{8}$ are congruent numbers. In
1975,  appealing to the Birch and Swinnerton-Dyer conjecture and
Shafarevich-Tate conjecture, Lagrange  \cite{serf1} deduced  a
conjecture on the parity of the $r(n)$  as follows:
$$  r(n) \equiv \left\{
  \begin{array}{ll}
  0 \ {\rm ( mod\ 2 ),}  & {\rm if } \  n \equiv 1,2,3 \ {\rm ( mod\ 8 );}
  \\
  1 \ {\rm ( mod\ 2 ),} &  {\rm if } \ n \equiv 5,6,7 \ {\rm ( mod\ 8 )}.
\end{array}
\right.
$$

\rm
The problem of constructing high rank CN-elliptic curves was considered by several authors.
In 1640, Fermat proved that $r(1)=0$, so $n=1$ is not a congruent
number. Billing \cite{biling} proved that $r(5)=1$.
Wiman \cite{wiman1} proved  that $r(34)= 2$, $r(1254)=3$ and $r(29274)=4$
\seqnum{A062693}, \seqnum{A062694}, \seqnum{A062695}  .
 In 2000, Rogers \cite{rgs1}, based on an idea of Rubin and Silverberg  \cite{rslb2}, found the first
 integers $n=4132814070,\ 61471349610$ such that $r(n)=5,6$, respectively.
 Later, in his PhD thesis \cite{rgs2},  Rogers
 gave  other integers with $r(n)=5,6$ smaller than those presented in \cite{rgs1}.
 Also he found \cite{rgs2} the first integer $n=797507543735$ with $r(n)=7$.
During the preparation of this paper, Rogers informed us that the
smallest $n$ with $r(n)=5$ which he was aware is $48272239$, while
the smallest $n$ with $r(n)=6$ is $6611719866$.
This rank $6$ curve is known to be minimal \cite{htt}.
Here we give the complete list on $n$'s with $r(n)=6$
communicated to us by Rogers \cite{rgs}, other than those curves
which are noted above:
\\ $66637403074$,  $94823967361$, $129448648329$, $179483163699$, $208645752554$,
$213691672290$,  \linebreak $226713842409$, $248767798521$,  $344731563386$,
  $670495125874$, $797804045274$, $898811499201$.

In Section 2,  we briefly describe the complete $2$-descents and $2$-Selmer rank
of CN-elliptic curves, denoted by $s(n)$, which is an upper bound
for $r(n)$. In Section 3, we describe Monsky's formula for computing
the value of $s(n)$. In Section 4, we study Mestre-Nagao's sum
method \cite{nag1, nag2, duje1} which is used as a sieving tool in
our algorithm. In Section 5, we design an algorithm to find high
rank CN-elliptic curves, based on  the Monsky's formula for
$2$-Selmer rank CN-elliptic curves $s(n)$, and Mestre-Nagao's sum
$S(N,n)$.
 We applied our algorithm for positive squarefree integers arisen from two specific families of congruent numbers.
We found a large number of curves  with rank $5$ and twenty-four
new curves with rank $6$. We have not found any new curve
with $r(n)\geq 7$, although with some variants of our method we
have rediscovered  Rogers' example with $r(n)=7$ (and some of
his examples with $r(n)=5$ and $6$). We have also found several
curves with $5\leq r(n)\leq 7$, where the upper bound is obtained
by MWRANK program (option {\tt -s}). It might be a challenging
problem to decide whether these curves have ranks equal to $5$ or $7$.

In our computations we used the PARI/GP
software (version 2.4.0) \cite{pari} and Cremona's  MWRANK program \cite{crem} for computing the
Mordell-Weil rank of the CN-elliptic curves (using the method of descent via 2-isogeny).

\section{ Complete {\rm 2}-descent and  {\rm 2}-Selmer rank   }

In this section, we briefly describe an upper bound for
Mordell-Weil rank of CN-elliptic curves $r(n)$, which is based on
the cardinality of $2$-Selmer group $S^{(2)}(E_n/\Q)$. We denote
this group by $S^{(2)}$.  For more details on the ($2$-)Selmer
groups and related topics, please see  \cite[Chap. X]{slm1}. In the
following we will describe $2$-descents  over $\Q$ for the
CN-elliptic curves. The number of $2$-descents is the order of
$S^{(2)}$. This is a power of $2$, and will be a multiple of $4$,
on account of the rational points of order $2$ on the curve $E_n$.
We  shall therefore write $\# S^{(2)}=2^{s(n)+2}$. The exponent
$s(n)$ is called  {\it $2$-Selmer rank}
of the curve $E_n$. Next we describe the $2$-descent
process on the curve $E_n$. For a similar argument of complete $2$-descent,  please see
\cite[Chap. X, \S 1]{slm1},  \cite[Sec. 3]{serf1} and \cite[Sec.
2]{heath1}.

Let $p_1$, $\dots$, $p_t$  be the odd prime factors of the squarefree integer $n$,
and let $M_{\Q}$ be the set of all places of $\Q$. Define the
sets $S$ and $\Q(S,2)$ as follows.
$$ S= \{ \infty , 2, p_1, \dots, p_t \}, $$
$$\Q(S,2)=\left \{a\in {\Q}^*/{{\Q}^*}^2 | \upsilon_p(a)\equiv 0\ ({\rm mod}\ 2)\
\forall p\in M_{\Q} \backslash S \right \}.$$

\begin{theorem}
\label{desc}
 Let $E_n$ be the elliptic curve $y^2=x^3-n^2x$ and let ${\cal O}$
 be the identity element of the group $E_n (\Q)$.
 With the above notation, we have:

\begin{description}
    \item[{\rm (i)}]
    There is an injective homomorphism
 $$\theta : E_n (\Q)/2E_n(\Q) \longrightarrow  \Q(S,2) \times \Q(S,2)$$

$$  P=(x,y) \mapsto \left \{
  \begin{array}{ll}
  (x, x-n),  & {\rm if } \ P \neq  {\cal O}, (0,0), (n,0);\\
  (-1, -n),  & {\rm if } \ P =  (0,0);\\
  (n, 2),  & {\rm if } \ P = (n,0);\\
  (1,1), &  {\rm if } \ P ={\cal O}.
\end{array}
\right.
$$
    \item[{\rm (ii)}]
    Let  $(a,b) \in \Q(S,2) \times \Q(S,2) \backslash \{(1,1),(-1,-n),(n,2) \}$.
    Then $(a,b)$ is the image of a point
$P=(x,y) \in E_n (\Q)/2E_n(\Q)$ if and only if the following
system of equations have a common solution
 $(X,Y,Z)  \in \Q^* \times \Q^* \times \Q^* $.
 $$(*) \ aX^2-bY^2=n, \ \ aX^2-abZ^2=-n.$$
If such a solution exist then one can take $P=(aX^2, abXYZ)=(bY^2+n, abXYZ)$.
\end{description}
\end{theorem}
For a proof of this theorem see \cite[Chap. X, \S 1]{slm1} or \cite[Sec. 3]{serf1} .

Note that the Mordell-Weil rank of the curve $E_n$ can be found by
 $$r(n)=\log_2\left ( \frac{Image (\theta)}{4}\right );$$
 Also, the cardinality of $S^{(2)}$ is equal to the number of the  pairs $(a,b)$ such that the system $(*)$ is  everywhere locally solvable.
 If one take the set
 $R=\left \{\pm 2^\alpha p_1^{\alpha_1} \cdots p_t^{\alpha_t} | \alpha, \alpha_1, \dots, \alpha_t \in \{0,1\} \right \}$
 as  representatives for $\Q (S,2)$, then it is immediate that $\#\Q (S,2)= 2^{t+2}$ and so
  $$r(n)\leq s(n)\leq 2 w(n).$$

\section{Monsky's formula for {\rm 2}-Selmer rank}

In 1994,  P. Monsky \cite{heath2} proved  a
theorem on the parity of the $2$-Selmer rank of CN-elliptic curves.
He  gave a formula for computation of the $s(n)$ through his proof
of this theorem.

\begin{theorem}
\label{mons}
 Let $n$ be a positive squarefree integer. Then
$$  s(n) \equiv \left\{
  \begin{array}{ll}
  0 \ {\rm ( mod\ 2 ),}  & {\rm if } \  n \equiv 1,2,3 \ {\rm ( mod\ 8 );}\\
  1 \ {\rm ( mod\ 2 ),} &  {\rm if } \ n \equiv 5,6,7 \ {\rm ( mod\ 8 )}.
\end{array}
\right.
$$
\end{theorem}
For a proof of this theorem see Appendix of
\cite{heath2}.

Let $n$ be a positive squarefree integer with odd prime factors
$p_1$, $\dots$, $p_t$. Define the diagonal
 $t \times t$ matrix $D_l=(d_i)$, for $l \in \{-1,-2,2\}$ , and the square $t \times t$ matrix
 $A=(a_{ij})$  as follows:
   $$
\begin{array}{ll}
$$d_i=\left\{
  \begin{array}{ll}
    0, & {\rm if} \ (\frac{l}{p_i})=1; \\
    1, & {\rm if} \ (\frac{l}{p_i})=-1,
\end{array}
\right.$$
 \ \
$$a_{ij}=\left\{
 \begin{array}{ll}
    0, & {\rm if} \ (\frac{p_j}{p_i})=1, j\neq i;  \\
    1, & {\rm if} \ (\frac{p_j}{p_i})=-1, j\neq i,
\end{array}\right.$$
\end{array}
\ \ a_{ii}=\sum_{j \,:\, j\neq i}^{} a_{ij}.$$
  Monsky showed that $s(n)$ can be computed as
 $$s(n)= \left\{
  \begin{array}{ll}
 2t- {\rm rank}_{\F_2} (M_o),  &{\rm if } \ n=p_1p_2\cdots p_t; \\
 2t- {\rm rank}_{\F_2} (M_{e}), &  {\rm if } \ n=2p_1p_2 \cdots p_t,
\end{array}
\right.$$ where $M_o$ and $M_e$ are the following $2t \times 2t$
matrices:
$$
\begin{array}{ll}
$$
 M_o=\left [
 \begin{array}{c|c}
  A+D_2 & D_2 \\
    \hline
    D_2 & A+D_{-2}
\end{array}
  \right],
  $$
&
$$  M_{e}=\left [
 \begin{array}{c|c}
 D_2 & A+D_2  \\
    \hline
 A^T+D_2 &  D_{-1}
\end{array}
  \right].$$
  \end{array}
  $$

 \section{Mestre-Nagao's sum} \rm

Now we describe a sieving method for finding the best candidates
for high rank CN-elliptic curves. For any  elliptic curve $E:
y^2=x^3+ax+b$ over $\Q$, and  every prime number $p$ not dividing
the discriminant $\Delta=-16(4a^3+27b^2)$ of $E$, we can reduce
$a$ and $b$ modulo $p$ and view $E$ as an elliptic curve over the
finite field $\F_p$. Let $\# E(\F_p)$ be the number of points on
the reduced curve:
$$\# E(\F_p)= 1+ \# \{0 \leq x,y\leq p-1 : y^2\equiv x^3+ax+b \ ({\rm mod}\  p)\}.$$
There is both theoretical and experimental evidence which suggests
that elliptic curves of high ranks have the property that $\# E(\F_p)$
is large for many primes $p$.

\begin{definition}\rm
  Let $N$ be a positive integer and ${\bf P}_N$ be the set of all primes less than $N$.
  Mestre-Nagao's sum is defined by
$$S(N,E)=\sum_{p\in {\bf P}_N}^{}(1- \frac{p-1}{\# E(\F_p)})\log p
=\sum_{p\in {\bf P}_N}^{} \frac{-a_p+ 2}{ \# E(\F_p)}\log p.$$
\end{definition}

Note that $S(N,E)$ can be computed efficiently with PARI/GP software \cite{pari}, provided $N$ is not too large.
  It is experimentally known  \cite{duje1,nag1,nag2} that we may
  expect that high rank curves have large $S(N,E)$.
  See \cite{camp} for a heuristic argument which connects this assertion with
  the famous Birch and Swinnerton-Dyer conjecture.
 For a positive squarefree integer $n$, we denote $S(N,E_n)$ by $S(N,n)$.

\section{An algorithm for finding  high rank }

Now we are ready to exhibit our algorithm for finding high rank
CN-elliptic curves, based on Monsky's formula for $2$-Selmer rank
of CN-elliptic curves $s(n)$ and Mestre-Nagao's sum $S(N,n)$. In
this algorithm, first of all, a list of different positive
squarefree congruent number is considered. Next, for any integer
$n$ in this list,  the value of $s(n)$  is computed by the
Monsky's formula which is described in the section 3. Selecting those
$n$'s with $s(n)\geq s$ for a given positive number $s$, a new
list of integers $n$ is scored by  Mestre-Nagao sum $S(N,n)$ using
finitely many successive primes. Finally, the Mordell-Weil rank
$r(n)$ is computed by MWRANK for integers $n$ with $s(n)\geq s$
and large values of Mestre-Nagao sums. To be more precise, we write
our algorithm step by step as follows.

\begin{description}
\item[Step 1.]  Let {\it s} be a positive integer. Choose a non-empty set $T$ of some squarefree congruent numbers.
               For any $n\in T$ compute   $s(n)$ by the Monsky's formula. Define the subset $T_s$ of $T$
                containing all $n \in T$ with $s(n)={\it s}$.  If   $T_s$ is empty  choose   another set
                $T$.
\item[Step 2.]
Let $k$ be a positive integer.
Choose the set ${\cal M}_s$ as follows:
               $${\cal M}_s=\left \{(N_i, M_i): \ 0 < N_1<\cdots <N_k , \ 0<M_i, \ 1\leq i \leq k \right\}.$$
             Put $ T_s^0=T_s$, and  for any $i$ with $1 \leq i\leq k$,
             define the recursive sets
             $$T_s^i=\left\{ n \in T_s^{i-1} : \ S(N_i,n)\geq M_i  \right\}.$$

\item[Step 3.]  Take $j$, $1\leq j \leq k$, such that for any $i$ with $j< i \leq k$, the sets $T_s^i$ are empty.
    Now for any $n \in T_s^j$, compute $r(n)$ using Cremona's MWRANK \cite{crem}.
\end{description}

\begin{remark}\rm
For a given positive integer {\it s} in Step 1, choice of
starting set $T$ is very important. To save the time, we should
avoid any repeated elements in $T$. By applying Theorem
\ref{mons} and  Lagrange's conjecture about the parity of $r(n)$,
one can expect to find an integer  $n$ in the set $T_s$ such that
$r(n)$ is less than $\it s$ and has the same parity as
{\it s}.\end{remark}

\begin{remark}\rm
The most sensitive part of our algorithm is choosing the sets
${\cal M}_s$ in Step 2. For a prescribed value of $s$, we must
choose the elements of ${\cal M}_s$ and its cardinality in such a
way that the total time of available computations is as small as
possible. Note that the elements of the sets $T_s^j$, in Step 3,
are the best candidates for high rank CN-elliptic curves.
\end{remark}

\begin{remark}\rm
In Step 3, we try to compute $r(n)$ for any $n \in T _s^j$.
This is done by  Cremona's program MWRANK efficiently for small
values of $n$. However, for large $n$'s the computation can be much slower,
and MWRANK often gives only lower and upper bounds for
$r(n)$.
\end{remark}

Given any positive integer $s$, our algorithm can be implemented in some  different
ways depending on the choice of the starting set $T$ in Step 1.
To explain our strategy, we need the next result which  gives two
specific families of congruent numbers.  For a proof of the cases
(I) and (II) see \cite{rob} and \cite{serf1}, respectively. Note
that the construction of congruent numbers via case (I) is the
same as that in \cite{rslb2} (originally due to $\rm
Gouv\acute{e}a$ and Mazur), applied to the curves $E_1:
y^2=x^3-x$ and $E_1^{'} : y^2=x^3+4x$.

 \begin{theorem}
 \label{tt2}
   Let $u$ and $v$ be arbitrary positive  integers such that $ u< v$,  ${\rm gcd }(u,v)=1$ and $u+v$ is odd.
   Then the squarefree parts of the following families of integers are congruent numbers:
 $${\rm (I)} \  uv(v-u)(v+u), \ \ {\rm (II)} \  uv(u^2+v^2)/2.  $$
 \end{theorem}

In this paper, we focused on the integers $s\geq 5$ and   all different positive squarefree integers $n$ of the forms (I) and (II)  with
$u<v \leq 10^5$ and $\omega(n) \geq 5$, where $\omega(n)$ denotes
the number of distinct prime factors of $n$.

After choosing  two sets $T_I$ and $T_{II}$ related to the integers of the form (I) and (II), we then took the starting set of the our algorithm as $T=T_I \cup T_{II}$ and
got different sets $T_s$ for each $s\geq 5$. Then  for each $s \ge 5$,
 we considered the related sets ${\cal M}_s$  as follows:
$$\{N_i\}_{i=1}^7=\left \{500,1000,5000,10000,15000,20000,50000  \right\}, \hspace{3cm}$$
$${\cal M}_5 =\left \{ (N_1,10),(N_2,12),(N_3,15),(N_4,20),(N_5,25),(N_6,28),(N_7,30) \right\},$$
$${\cal M}_6 =\left \{ (N_1,10),(N_2,14),(N_3,18),(N_4,22),(N_5,25),(N_6,30),(N_7,35) \right\},$$
$${\cal M}_7 =\left \{ (N_1,10),(N_2,15),(N_3,20),(N_4,25),(N_5,30),(N_6,35),(N_7,40) \right\},$$
$${\cal M}_8 =\left \{ (N_1,10),(N_2,14),(N_3,16),(N_4,20),(N_5,25),(N_6,30),(N_7,35) \right\},$$
$${\cal M}_9 =\left \{ (N_1,10),(N_2,15),(N_3,20),(N_4,25),(N_5,28),(N_6,30),(N_7,35) \right\},$$
$${\cal M}_{\geq 10} =\left \{(N_1,10),(N_2,12),(N_3,15),(N_4,18),(N_5,22),(N_6,25),(N_7,30) \right\}.$$
 For each $s \ge 5$ and each $i$, $1 \leq i \leq 7$, by choosing $(N,M)=(N_i,M_i)\in {\cal M}_s$
 and computing $S(N_i,n)$ for all $n\in T_s^{i-1}$,  gets  the
 sets $T_s^i$  of $n$'s that satisfy $S(N_i,n) \geq M_i$.
 The elements of the sets $T_s^j$ are best
 candidates to give high rank CN-elliptic curves.
Finally, we used MWRANK to compute Mordell-Weil rank $r(n)$, for
$n$'s in each of the sets $T_s^j$. This stage of our algorithm
was very time consuming. By the implementation of our algorithm,
we have rediscovered some of the Rogers' examples with
$r(n)=5, 6,$ and $ 7$. Also, we were able to find some new
CN-elliptic curves with $r(n)=6$ and some curves with $5 \leq
r(n) \leq 7$. We give these curves in the Tables 1 and 2,
respectively.

We give also generators of the Mordell-Weil group for two smallest new examples with $r(n)=6$.
By using MWRANK we find 6 indepenent points on $E_n$, which are moreover generators of
the Mordell-Weils group, while LLL-algorithm is used for finding the generators with smaller
heights, which are listed below.

For
$n=531670544130$ we have the curve
$$ y^2 = x^3-282673567495490277456900x $$
with the generators
\small
\begin{verbatim}
P1 = [-317205078080, 240309412570889200],
P2 = [1110744023070, 1027815645288207600],
P3 = [-8842721250, 49989119984694000],
P4 = [2350922039070, 3511212519485048400],
P5 = [7424745951989070/361, 639554031769152257946000/6859],
P6 = [-165395800834700271/51351556, 11103259191546833925683935833/367985250296]
 \end{verbatim}
  \normalsize

For $ n=602730488666$
we have the curve
$$ y^2 = x^3-363284041967555154459556x $$
with the generators
\small
\begin{verbatim}
P1 = [25844642800106/25, 106746067884077780496/125],
P2 = [-89776938384, 178580334935648520],
P3 = [3666632085466, 6925523273366507040],
P4 = [26198594092166458/10609, 4112253205326835858960032/1092727],
P5 = [2097707297289652801/1012036, 2906919721960250194451760705/1018108216],
P6 = [5187004732864967512122/8543489761,
     44888914750852091711316911386224/789683302098991]
\end{verbatim}
\normalsize

\vfill

\begin{table}[hbp!]
\label{tab3}
\begin{center}
{\footnotesize
\begin{tabular}{|l|l|c|c|}
\hline
$n$ & factorization &  $n \bmod{8}$   &   $s(n)$ \\
\hline
 531670544130         & 2\cd3\cd5\cd11\cd17\cd107\cd463\cd1913                       & 2  & 6 \\
 602730488666         & 2\cd29\cd41\cd97\cd137\cd19073                               & 2  & 6  \\
 1079812755065        & 5\cd11\cd23\cd41\cd89\cd449\cd 521                           & 1  & 6  \\
 1351528542210        & 2\cd3\cd5\cd7\cd11\cd29\cd31\cd47\cd61\cd227                 & 2  & 6 \\
 1440993982946        & 2\cd7\cd17\cd23\cd41\cd73\cd281\cd313                        & 2  & 8  \\
 1544991154746        & 2\cd3\cd13\cd19\cd83\cd163\cd251\cd307                       & 2  & 6  \\
 1663586838899        & 17\cd103\cd137\cd756\cd9161                                  & 3  & 8  \\
 2280190889130        & 2\cd3\cd5\cd7\cd11\cd23\cd41\cd257\cd4073                    & 2  & 6  \\
 4611082954146        & 2\cd3\cd19\cd41\cd113\cd953\cd9161                           & 2  & 8  \\
 8231905771386        & 2\cd3\cd11\cd17\cd19\cd23\cd41\cd43\cd89\cd107               & 2  & 6  \\
 9033322597530        & 2\cd3\cd5\cd7\cd11\cd43\cd53\cd59\cd127\cd229                & 2  & 6 \\
 17434310103210       & 2\cd3\cd5\cd7\cd11\cd13\cd17\cd19\cd67\cd139\cd193           & 2  & 6  \\
 46485304142530       & 2\cd5\cd11\cd19\cd23\cd43\cd67\cd107\cd3137                  & 2  & 6  \\
 90181020280890       & 2\cd3\cd5\cd7\cd11\cd251\cd397\cd401\cd977                   & 2  & 6  \\
 165130972136130      & 2\cd3\cd5\cd7\cd11\cd13\cd29\cd103\cd233\cd7901              & 2  & 6  \\
 179009302343970      & 2\cd3\cd5\cd7\cd17\cd19\cd23\cd47\cd53\cd73\cd631            & 2  & 6 \\
 181025271456226      & 2\cd17\cd103\cd127\cd151\cd1259\cd2141                       & 2  & 6 \\
 243339180933145      & 5\cd11\cd401\cd1049\cd3169\cd3319                            & 1  & 8  \\
 339507119347242      & 2\cd3\cd7\cd17\cd19\cd23\cd37\cd59\cd113\cd401               & 2  & 6  \\
 444724421083665      & 3\cd5\cd17\cd31\cd71\cd103\cd137\cd233\cd241                 & 1  & 8  \\
 846249312638730      & 2\cd3\cd5\cd7\cd11\cd13\cd31\cd37\cd41\cd101\cd349           & 2  & 6  \\
 1056710141801930     & 2\cd5\cd7\cd11\cd41\cd43\cd53\cd71\cd269\cd769               & 2  & 6  \\
 4601440550332626     & 2\cd3\cd7\cd11\cd13\cd17\cd19\cd37\cd41\cd101\cd113\cd137    & 2  & 6  \\
 13897395819317010    & 2\cd3\cd5\cd7\cd11\cd13\cd23\cd29\cd31\cd61\cd113\cd191      & 2  & 6  \\
\hline
\end{tabular}
}
\end{center}
\caption{\small Some new CN-elliptic curves with $r(n)=6$}
\end{table}
\normalsize

\begin{table}[hbp!]
\label{tab4}
\begin{center}
{\footnotesize
\begin{tabular}{|l|l|c|c|}
\hline
$n$ & factorization &  $n \bmod{8}$   &   $s(n)$  \\
\hline
 1024801887174       & 2\cd3\cd13\cd37\cd409\cd769\cd1129                  & 6 & 7  \\
 1025774078934       & 2\cd3\cd11\cd17\cd41\cd43\cd641\cd809               & 6 & 7 \\
 1649085975174       & 2\cd3\cd11\cd47\cd73\cd97\cd193\cd389               & 6 & 7 \\
 2093383150230       & 2\cd3\cd5\cd29\cd73\cd97\cd419\cd811                & 6 & 7 \\
 2392760979654       & 2\cd3\cd17\cd41\cd43\cd83\cd160313                  & 6 & 7  \\
 2473595024934       & 2\cd3\cd11\cd17\cd41\cd83\cd347\cd1867              & 6 & 7  \\
 5080701332454       & 2\cd3\cd11\cd17\cd41\cd59\cd521\cd3593              & 6 & 7  \\
 5449406258406       & 2\cd3\cd11\cd17\cd41\cd251\cd683\cd691              & 6 & 7  \\
 7322494848870       & 2\cd3\cd5\cd17\cd19\cd137\cd151\cd36529              & 6 & 7  \\
 7391341307526       & 2\cd3\cd11\cd19\cd59\cd67\cd523\cd2851              & 6 & 7  \\
 7697325362694       & 2\cd3\cd11\cd137\cd401\cd547\cd3881                 & 6 & 7  \\
 7836495180886       & 2\cd17\cd281\cd353\cd971\cd2393                     & 6 & 9  \\
 7889458857566       & 2\cd11\cd19\cd881\cd1049\cd1571                     & 6 & 7 \\
 8549294440966       & 2\cd17\cd19\cd37\cd137\cd353\cd5857                 & 6 & 7  \\
 10571147972390      & 2\cd5\cd17\cd89\cd277\cd587\cd4297                  & 6 & 7  \\
 11050024116846      & 2\cd3\cd11\cd13\cd17\cd29\cd31\cd569\cd1481         & 6 & 7  \\
 12651761296614      & 2\cd3\cd11\cd17\cd19\cd43\cd59\cd449\cd521          & 6 & 7  \\
 14020765617254      & 2\cd11\cd17\cd23\cd71\cd241\cd 95257                & 6 & 7  \\
 19843964725254      & 2\cd3\cd17\cd19\cd937\cd2683\cd4073                 & 6 & 7  \\
 25161173711039      & 19\cd23\cd29\cd103\cd1657\cd11633                   & 7 & 7  \\
 25837148295902      & 2\cd31\cd97\cd593\cd1217\cd5953                     & 6 & 9 \\
 26755379766174      & 2\cd3\cd23\cd59\cd233\cd353\cd39953                 & 6 & 7   \\
 29130582949206      & 2\cd3\cd19\cd113\cd283\cd1913\cd4177                & 6 & 7   \\
 32334652741974      & 2\cd3\cd11\cd43\cd89\cd113\cd883\cd1283             & 6 & 7   \\
 34243576397574      & 2\cd3\cd73\cd89\cd457\cd953\cd2017                  & 6 & 7  \\
 35876712238310      & 2\cd5\cd31\cd41\cd1289\cd1361\cd1609                & 6 & 7  \\
 44066140293846      & 2\cd3\cd11\cd17\cd41\cd43\cd59\cd491\cd769          & 6 & 9  \\
 56858065281654      & 2\cd3\cd7\cd13\cd19\cd73\cd89\cd769\cd 1097         & 6 & 7 \\
 57705905931141      & 3\cd13\cd17\cd131\cd521\cd937\cd1361                & 5 & 7 \\
 57939619068870      & 2\cd3\cd5\cd7\cd11\cd37\cd53\cd89\cd137\cd1049      & 6 & 7  \\
 61639096639029      & 3\cd7\cd13\cd29\cd241\cd2113\cd15289                & 5 & 7 \\
 109995988504269     & 3\cd17\cd41\cd65809\cd114193                        & 5 & 7 \\
 114490690064454     & 2\cd3\cd11\cd19\cd577\cd1873\cd 84481               & 6 & 9  \\
 117205364344206     & 2\cd3\cd7\cd17\cd73\cd97\cd233\cd293\cd2377         & 6 & 7  \\
 119231629856526     & 2\cd3\cd11\cd17\cd29\cd41\cd59\cd83\cd18251         & 6 & 7  \\
 121466637600990     & 2\cd3\cd5\cd11\cd17\cd31\cd89\cd107\cd1033          & 6 & 7  \\
 130629627999390     & 2\cd3\cd5\cd13\cd17\cd37\cd41\cd97\cd257\cd521      & 6 & 7  \\
 146421396607926     & 2\cd3\cd11\cd17\cd19\cd449\cd2417\cd6329            & 6 & 7  \\
 175656508365734     & 2\cd11\cd97\cd113\cd10169\cd71633                   & 6 &  9 \\
 180196195115046     & 2\cd3\cd11\cd17\cd43\cd83\cd179\cd251393            & 6 &  7 \\
 191519081464326     & 2\cd3\cd7\cd11\cd31\cd41\cd59\cd89\cd89\cd179\cd347   & 6 &  7 \\
 242515586992326     & 2\cd3\cd19\cd41\cd73\cd587\cd641\cd1889             & 6 &  9 \\
 433182183087126     & 2\cd3\cd11\cd17\cd41\cd251\cd2707\cd13859           & 6 & 7  \\
 459848288031405     & 3\cd5\cd7\cd13\cd17\cd41\cd61\cd389\cd20369         & 5 & 7  \\
 1687029282320910    & 2\cd3\cd5\cd11\cd1049\cd1729\cd2027                 & 6 & 7  \\
 2053424339679966    & 2\cd3\cd11\cd17\cd19\cd31\cd43\cd179\cd499\cd809    & 6 & 7  \\
 2059195525185430    & 2\cd5\cd89\cd641\cd823\cd929\cd4721                 & 6 & 9\\
 3167344617712806    & 2\cd3\cd19\cd73\cd89\cd283\cd3137\cd4817            & 6 & 9 \\
 8797235243700486    & 2\cd3\cd11\cd19\cd313\cd577\cd5147\cd7547           & 6 & 9  \\
 342916139097905191  & 3\cd13\cd17\cd37\cd53\cd61\cd157\cd1753\cd6733      & 7 & 7  \\
\hline
\end{tabular}
}\vspace{-2ex}
\end{center}
\caption{\small Some CN-elliptic curves  with $5 \leq r(n)\leq 7$}
\end{table}

\normalsize

\section{Acknowledgements}
The authors would like to express their gratitude to N. Rogers
for giving the list  of his unpublished results. They also thank
the referee for his/her helpful suggestions. The third author
would like to thank J. Cremona for his helpful guides for using
 MWRANK and his suggestions to resolve certain
 computational issues in computing ranks of CN-elliptic curves for large positive integers.


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\newblock {}{\small
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\end{thebibliography}

\bigskip
\hrule
\bigskip

\noindent 2000 {\it Mathematics Subject Classification}:
Primary 11G05; Secondary 14H52.

\noindent \emph{Keywords: } 
CN-elliptic curve, Mordell-Weil rank, $2$-Selmer rank, Mestre-Nagao sum.

\bigskip
\hrule
\bigskip

\noindent
(Concerned with sequences
\seqnum{A003273},
\seqnum{A006991},
\seqnum{A062693},
\seqnum{A062694}, and
\seqnum{A062695}.)

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received April 5 2009;
revised version received  July 14 2009.
Published in {\it Journal of Integer Sequences}, July 16 2009.

\bigskip
\hrule
\bigskip

\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}.
\vskip .1in

\end{document}

.
