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\title{Positive Rank Examples of the \\ Conjecture of Birch and Swinnerton-Dyer}
\author{ Francis Ois\'{\i}n McGuinness\\
         Department of Mathematics \\
          Fordham University\\
         Bronx NY 10458}

\date{January 23 1989}

\def\legendre{\overwithdelims()} % Use as {a \legendre b } See T. p152
\def\pmod#1{\allowbreak\mkern9mu({\rm mod}\,\,#1)}   % my pmod
\newcommand{\SwD}{Swinnerton-Dyer}
\newcommand{\BS}{Birch-\SwD}
\newcommand{\Q}{{\bf Q}}
\newcommand{\Z}{{\bf Z}}
\newcommand{\R}{{\bf R}}
\newcommand{\C}{{\bf C}}
\renewcommand{\H}{{\cal H}}
\newcommand{\isom}{\cong}
\newcommand{\Gal}{\mbox{\rm Gal\,}}
\newcommand{\Aut}{\mbox{\rm Aut\,}}
\newcommand{\rank}{\mbox{\rm rank\,}}
\newcommand{\hgtp}[2]{\langle #1, #2 \rangle}
\newcommand{\hgt}[1]{\hgtp{#1}{#1}}
\newcommand{\F}[1]{{\bf F}_{#1}} % use in math mode only
\newcommand{\Fl}{\F{l}}
\newcommand{\Fls}{\F{l^2}}

\newcommand{\Ls}{L(E,s)}
\newcommand{\Sha}{\hbox{$\amalg\kern-.36em\amalg$}} 
\newcommand{\divides}{{\,\vert\,}}      
\newcommand{\notdivides}{{\!\not\vert\,}}


\newcommand{\beq}{\begin{equation}}
\newcommand{\eeq}{\end{equation}}

\newtheorem{lemma}{Lemma}[section] % see LaTeX book page 58 for more
\newtheorem{theorem}[lemma]{Theorem} % see LaTeX book page 58 for more

\begin{document}
\maketitle
\begin{abstract}
We describe some computations related to some rank 1 and rank 2 cases of the
conjecture of Birch and \SwD. Appendices describe height and period computations
on elliptic curves.
\end{abstract}

\section*{Introduction} Suppose that $E$ is an elliptic curve defined over \Q.  Then the
conjecture of Birch and \SwD\  states that the Hasse-Weil $L$-series $\Ls$ has an
analytic continuation to an entire function on \C, with a zero at $s=1$ of order $r$, the
Mordell-Weil rank of $E$  over $\Q$ (=free $\Z$-rank of the finitely generated abelian group
$E(\Q)$.). The conjecture also predicts the leading term of the Taylor series of $\Ls$ at $s=1$.
Namely, \[
\lim_{s\to1}\frac{\Ls}{(s-1)^r} = 
\Omega \frac{|\Sha|\det\langle P_i, P_j\rangle}{[E(\Q):E']^2}\prod_{p\divides\Delta}c_p.
\]
We recall briefly (see \cite{Tate} or \cite{Silvb} for more details) that
$\Omega$ denotes $\int_{E(\R)}|\omega|$ where $\omega$ is a N\'eron differential
on $E$, so that $\Omega$ is $[E(\R):E_0(\R)]$ times a fundamental real period of $E$,
$\Sha$ denotes the (conjecturally finite) Tate-Shafarevich group of $E$, the $P_i$ for
$1\leq i\leq r$ are an independent set of points in $E(\Q)$ generating the subgroup
$E'$, and $\hgtp{P_i}{P_j}$ denotes the height pairing. Finally $\Delta$ is the minimal
discriminant of $E$, and the fudge factors $c_p=[E(\Q_p):E_0(\Q_p)]$, where
$E_0(\Q_p)$ denotes the connected component of the group of $p$-adic points on $E$.

 In this paper we discuss the numerical verification of some
rank~1 and rank~2 examples of the conjecture. Ligozat (see \cite{Lig})
has verified many rank~0 examples when $E$ is an elliptic modular curve,
up to an assumption about the Tate-Shafarevich group. Other rank~0 examples
for curves with complex multiplication have been explored by Birch and
\SwD \cite{BSwD1} and by Razar\cite{Raz1,Raz2}, while Stephens \cite{Steph}has
given some complex multiplication examples with rank~1.  A rank~3 example of
particular interest is described in \cite{BuGrZa}. 
Our main examples will be quotients
of modular curves. Since this work was originally done, the work of Gross and Zagier
(\cite{GrZa}) has shown that, in our examples, the  numerical approximate equality we
describe is an actual equality, up to an assumption about the Tate-Shafarevich group.
Furthermore, Rubin  (\cite{Rubin}) has proved the finiteness of the Tate-Shafarevich
group for curves with complex multiplication.

The first section discusses the rank~1 examples,
section~2 discusses a rank~2 example, while the remaining sections
(which are in the nature of appendices) describe how the computations of the
period, heights and $L$-series derivative values
were performed.

The work described in this paper was carried out at the University
of Virginia, and made use of its computational facilities.

Some frequent notation: $|S|$ will denote the size of a finite set $S$. Most of the
notation is quite standard, and can be found in \cite{Silvb}, for example.

\section{Some Rank 1 Curves with Prime Conductor}
The first curve (i.e., the elliptic curve over $\Q$ of least
conductor) with positive rank is the curve
\[
37A \qquad y^2+y=x^3-x\qquad \Delta=N=37,
\]
for which $P=(0,0)$ generates $E(\Q)\isom\Z$. In fact, this is the first
of the nine elliptic curves of the form $X_0(p)/w$, corresponding
to $p=37$, 43, 53, 61, 79, 83, 89, 101 and 131.  Recall that
$X_0(p)$ is the modular curve of level~$p$ that classifies
pairs $(E,C)$ where $E$ is an elliptic curve and $C$ is a cyclic subgroup
of order~$p$. The complex points of $X_0(p)$ correspond to
$\H^{*}\backslash\Gamma_0(p)$ where $\H^{*}$ denotes the upper half
plane together with the cusps, and 
\[
\Gamma_0(p) = \left\{ \left(\begin{array}{cc} a & b \\ c &d\end{array} \right)
: ad-bc=1, a,b,c,d\in\Z , c\equiv0\bmod{p}\right\}.
\]
The curve $X_0(p)$ is defined over $\Q$, and the canonical
involution $w=w_p$ (induced from the map on $\H$ given
by $z\mapsto-1/pz$ or by the map on moduli $(E,C)\mapsto (E/C, E[p]/C)$),
acts $\Q$-rationally so that the quotient $X_0(p)/w$ is also
a curve defined over $\Q$.  There are well-known formulas for
the genus of both $X_0(p)$ and its quotient $X_0(p)/w$. We first
show that the quotient curve is of genus~1 for exactly the
values of $p$ given above.
\begin{lemma}
$X_0(p)/w$ has genus~1 if and only if $p=37$, 
43, 53, 61, 79, 83, 89, 101 or 131.
\end{lemma}
To prove this we apply the technique described in section~5 of
\cite{Ogg:pspm37}. (A more abstract formulation can be found in
\cite{Mazur:Eis}, page~146, Lemma~2.6.) Denote $X_0(p)$ by $X$ and $X_0(p)/w$ by
$\bar{X}$,  and consider the double cover map  \[ 
X \to \bar{X}.
\]
For a prime $l\neq p$, we can view this map over $\Fl$, and estimate
the number of $\Fls$ points on $X$. On the one hand, the supersingular
points of $X$ (those points classifying supersingular elliptic curves) are
defined over $\Fls$, and we get
\[
\left|X(\Fls)\right| \geq 2 + \frac{(l-1)(p+1)}{12},
\]
where 2 is the number of cusps of $X$. On the other hand, we have
\begin{eqnarray*}
\left|X(\Fls)\right| & \leq  & 2\left|\bar{X}(\Fls)\right| \\
          & \leq  & 2(1 + 2\bar{g}l + l^2),
\end{eqnarray*}
using the Weil bound for the number of points on $\bar{X}$ defined over
$\Fls$, where $\bar{g}$ is the
genus of $\bar{X}$. Taking $l=2$ and combining these two inequalities yields
the bound on $\bar{g}$:
\[
\bar{g} +1 \geq \frac{p+1}{96},
\]
which for $\bar{g}=1$ yields $p\leq 191$. A direct computation using
the genus formula for $\bar{g}$ shows that the only primes less than~191
for which the genus is~1 are those listed.

For completeness, we state the genus formula used in the computation just 
mentioned.  If $g$ denotes the genus of $X$, then we have
\[
g = \frac{1}{12}(p-\nu),
\]
where $p\equiv \nu\pmod{12}$, $\nu\in\{13,5,7,-1\}$, and
\[
\bar{g} = \frac{1}{4}(2g+2-f),
\]
from the Riemann-Hurwitz formula, where $f$ is the number
of fixed points of $w$ acting on $X$. Since $w$ interchanges the two cusps
$0$ and $\infty$ of $X=X_0(p)$, $f$ is the number of fixed points
of $w$ on the open modular curve $Y_0(p)$, and can be computed
using the modular interpretation of $Y_0(p)$. The result is (see 
\cite{Ogg:Hyper} for the details) that $f=h$, $2h$ or $4h$ depending
on whether $p\equiv 1\pmod{4}$, $p\equiv7\pmod{8}$ or
$p\equiv3\pmod{8}$ respectively, and where $h=h(-p)$ is the class number
of the imaginary quadratic field $\Q(\sqrt{-p})$. (We could use
the standard upper bounds on $h$ to bound $\bar{g}$ from below, but
the bound on $p$ thus obtained when $\bar{g}=1$ is much larger than that obtained
from the argument already given.)

For the rest of this section, we let $E$ denote $X_0(p)/w$ for one
of the nine primes listed in Lemma~(1.1) above. The two cusps of $X_0(p)$ map to a
rational point on $E$ which is therefore an elliptic curve defined over $\Q$. Our
immediate task is to find the discriminant and conductor of $E$. Firstly, by the results
of  Igusa-Deligne-Rapaport on the reduction properties of $X_0(p)$, $E$ has
good reduction at all primes $l\neq p$. Then by the determination of the fibre
$J_p$ at $p$ of the N\'eron model of the Jacobian of $X_0(p)$ (as given in
\cite{Mazur:Eis}, appendix), it follows that $E$ has multiplicative reduction
at~$p$. (The connected component $J_p^0$ is a group of multiplicative
type of which the fibre at~$p$ of the N\'eron model of $E$ is a quotient.)
Furthermore, the fibre at~$p$ of $E$ has one irreducible component, since
the quotient $J_p/J_p^0$ is cylic generated by the class of the divisor
$(0)-(\infty)$, and this divisor is killed by the action of $w$. We conclude
that $E$ has conductor~$p$, and discriminant $\Delta=\pm p$. (See the table on
page~46 of \cite{Mod4} for the last conclusion.)

Next we find a model for $E$ by examining the table of curves of conductor up to
200 given in \cite{Mod4}. For the primes $p=43$, 53, 61, 79, 83, 101 and 131, there is
only one curve of conductor~$p$ in the table, and this curve must be $E$.
For $p=37$ there are three curves, and for $p=89$ there are two curves
to be considered. We can identify $E$ by noting that for a Weil curve of
conductor~$p$, the eigenvalue of the action of $w$ on the corresponding
newform is $-a_p$ in the usual notation. But for $E$, this eigenvalue
must be~1, and so we have $a_p=-1$. This shows that when $p=37$,
we have $E=37A$, and when $p=89$, we have $E=89C$, since the other curves
have $a_p=1$.\footnote{The table
header to table~3 in \cite{Mod4} is misleading here.}
 Recall that the geometric significance of $a_p=-1$ is that the tangents
to the node of $\tilde{E}$ ($E$ reduced modulo~$p$) are not $\F{p}$ rational.

It follows from the table that the curves $E$ that we are considering are 
unique in their isogeny class. This is also a consequence of  some results of Serre,
see pages~306--308 and (5.5.6)--(5.5.8) of \cite{Serre}. If 
$\phi_l:\Gal(\bar{\Q}/\Q) \to \Aut E[l] \isom GL_2(\Fl)$ denotes
the Galois representation on the $l$-torsion points of $E$, then the corollary
to Proposition~21 in \cite{Serre} (take $p=2$) shows that $\phi_l$ is surjective
for all $l>5$, and then computing $|E(\Fl)|$ for $l=2$, 3 (for $p=61$ and
101 it is necessary to look at $l=5$ also), shows that $\phi_l$ is surjective
for all primes~$l$. This also implies that $E(\Q)$ has no torsion. Since
a non-trivial rational point is listed in the tables for each of our
curves $E$, we must have that $\rank E(\Q)\geq 1$. The 2-descent bounds
in \cite{Br-Kr} then show that $E$ has rank~1 (and that the 2-torsion in
the Tate-Shafarevich group of $E$ is zero). In the table below I give
for reference the coefficients of a defining equation for $E$, the sign of 
$\Delta$, and the coordinates of a point $P$ of infinite order.
\[
\begin{tabular}{|r|c|c|c|c|c|c|c|}
\hline
Curve & $a_1$ & $a_2$ & $a_3$ & $a_4$ & $a_6$ & sign $\Delta$ & $P$ \\
\hline
37A & 0 & 0 & 1 & $-1$ & 0 & $+$ & (0,0) \\
43A & 0 & 1 & 1 & $0$ & 0 & $-$ & (0,0) \\
53A & 1 & $-1$ & 1 & 0 & 0 & $-$ & (0,0) \\
61A & 1 & 0 & 0 & $-2$ & 1 & $-$ & (1,0) \\
79A & 1 & 1 & 1 & $-2$ & 0 & $+$ & (0,0) \\
83A & 1 & 1 & 1 & 1 & 0 & $-$ & (0,0) \\
89C & 1 & 1 & 1 & $-1$ & 0 & $-$ & (0,0) \\
101A & 0 & 1 & 1 & $-1$ & $-1$ & $+$ & ($-1$,0) \\
131A & 0 & $-1$ & 1 & 1 & 0 & $-$ & (0,0) \\ \hline
\end{tabular}
\]
In each case $P$ actually generates $E(\Q)$. This can be verified by
analysing the standard proof of the Mordell-Weil theorem. We omit
the details. We remark that the theory of Heegner points
provides an intrinsic way to exhibit points on $E$, without making reference
to a defining equation. For example, \cite{Mazur:Eis} page~147
shows that $E$ has positive rank using Heegner points. As specific examples 
we mention that Gross (private
communication) has shown that the quadratic field $\Q(\sqrt{-67})$
yields the Heegner point $P=(0,0)$ on 37A, while $\Q(\sqrt{-115})$
yields the point $6P$. We summarize the main results so far.
\begin{theorem} For the nine values of $p$ in
$\{37, 43, 53, 61, 79, 83, 89, 101, 131\}$, the curve
$X_0(p)/w$ is  an elliptic curve, defined over $\Q$,
of conductor $p$, and
with $E(\Q)\isom \Z$.
\end{theorem}

Now that we have the curves $E$ of rank~1 to hand, let us restate the conjecture
of Birch and Swinnerton-Dyer specialised to this situation. Noting
that $c_p=1$ (see \cite{Mod4} page~46, $E$ has reduction type $I_1$ at $p$), we have
the statement 
\begin{center} 
$L(E,s)$ has a zero of order~1 at $s=1$ and
$L'(E,1)=\Omega\left|\Sha\right|\hgt{P}$.
 \end{center}
We describe the calculations that corroborate this statement. Firstly,
computing the period $\Omega$ is described in the first
appendix below. It can be done very rapidly and accurately using Gauss'
Arithmetic-Geometric Mean. Next we describe how the height is computed.
We have $\hgt{P}=2h^{*}(P)$ where $h^{*}(P)$ is the N\'eron-Tate canonical height
of a point $P$ on an elliptic curve.
As recalled below, the canonical height function is a sum of
local heights,
\[
h^{*}(P) =\sum_v\lambda_v(P),
\]
where the sum is taken over all places $v$ of $\Q$. For the finite places $v$ Tate
has given explicit formulas for the local heights $\lambda_v(P)$, and has
given a rapidly converging algorithm to compute $\lambda_{\infty}(P)$ for
points $P\in E(\R)$, see
\cite{Silv} for the details (and an improvement of Tate's algorithm to
cope with complex points also). In our cases, where $E$ and $P$ are as given,
since $P$ reduces to a non-singular point modulo {\em every\/} prime, it
follows that
\[
\lambda_v(P)=0
\]
for all {\em finite\/} places $v$ of $\Q$. Then $h^{*}(P)=\lambda_{\infty}(P)$
is easily calculated. 
(We give the algorithm in the section on heights below.)

It remains  to compute $L'(E,1)$. Since $E=X_0(p)/w$,
the Hasse-Weil $L$-series $\Ls$ of $E$ is the Mellin transform of a newform
$f$ of weight~2 for the group $\Gamma_0(p)$ which is invariant under $w$, $f|w=f$.
The functional equation for $\Ls$ takes the form that
\[
\Phi(s)=(2\pi)^{-s}\Gamma(s)\Ls =\int_0^{\infty}t^{s-1}f(it)\,dt
\]
satisfies
\[
\Phi(s)=-p^{1-s}\Phi(2-s).
\]
We recall that the functional equation just stated can be proved by
splitting the integral from~0 to $\infty$ at $p^{-1/2}$ (where
$w$ acting on $\H$ fixes $ip^{-1/2}$), and then using $f\vert w=f$ to
obtain the representation
\[
\Phi(s) = \int_{p^{-1/2}}^{\infty} (t^{s-1}-p^{1-s}t^{1-s})f(it)\,dt
\]
which is valid for all $s\in\C$ and which provides the analytic continuation
of $\Ls$ to an entire function on $\C$. We also see from the functional
equation that $L(E,1)=0$ and that $\Ls$ has an odd-order zero at $s=1$.
Suppose that we write 
\[
f(z)=\sum_{n=1}^{\infty}a_nq^n,
\]
where as usual $q=e^{2\pi iz}$, then differentiating the last expression
for $\Phi(s)$ once with respect to $s$, setting $s=1$ and integrating
the series for $f$ term by term we get
\[
L'(E,1) = 2\sum_{n=1}^{\infty}\frac{a_n}{n}
\int_{2\pi n/\sqrt{p}}^{\infty} e^{-x}\,\frac{dx}{x}.
\]
The coefficients $a_n$ have the multiplicativity property
expressed by the existence of an Euler product of the expected
form for $\Ls$ (or alternatively by the fact that $f$ is a newform, and
therefore an eigenfunction of the Hecke operators). We have
$a_1=1$, $a_p=-1$ and $a_l=1+l-|E(\Fl)|$ for $l\neq p$ a prime. Both the sum
and the integrals in our formula for $L'(E,1)$ converge rapidly and a moderate
number of terms is enough to obtain accurate values. Standard estimates
show that taking the terms for $n\leq 60$ and truncating the 
integrals at~$x=64$ gives $L'(E,1)$ correct to 11 decimal places. 
(See the third appendix for more details.)
A convenient method to calculate the $a_l$  (which can also be found
in Table~3 in \cite{Mod4}) for $l>2$ is to use
\[
a_l = -\sum_{x\bmod l}{f(x)\legendre l},
\]
where $f(x)=x^3+b_2/4x^2+b_4x/2+b_6/4$, and where the $b_i$ are calculated from
the coefficients of the minimal equation for $E$ according to Tate's 
formulaire. 

Finally, carrying out all the computations results in the numbers in the table
below. The predicted and calculated values for $L'(E,1)$ agree to at least 8 
places in each case, so that the predicted order for the Tate-Shafarevich 
group for each of our nine curves is~1.
\[
\begin{tabular}{|r|l|l|l|l|}
\hline
Curve & $\Omega$ & $h^{*}(P)$ & $\Omega\hgt{P}$ & $L'(E,1)$ \\ \hline
37A & 5.986917292 & .025555704 & .3059997736 & .3059997738 \\
43A & 5.468689531 & .031408754 & .3435239797 & .3435239746 \\
53A & 4.687641049 & .046490742 & .4358638242 & .4358638241 \\
61A & 6.133193148 & .039593865 & .4856736508 & .4856736514 \\
79A & 5.950800353 & .048832105 & .5811802165 & .5811802159 \\
83A & 3.374468900 & .088646147 & .5982673327 & .5982673327 \\
89C & 5.552626565 & .056052440 & .6224765415 & .6224765415 \\
101A& 4.590247212 & .082351726 & .7560295657 & .7560295656 \\
131A& 4.171609276 & .108047599 & .9014647353 & .9014647353 \\ \hline
\end{tabular}
\]
There are two other curves with prime conductor and positive rank in
the tables in \cite{Mod4}, namely the curves 163A and 197A. Similar
computations can be carried out in these cases to check
that $L'(E,1)=\Omega\hgt{P}$ (at least approximately). The formula
for $L'(E,1)$ of course assumes that $E$ is a Weil curve, which we
haven't verified in these two cases. For the record, the resulting
values of $L'(E,1)$ are 1.04793302 and .78732047 
respectively.\footnote{All the period and height calculations were
carried out on a Texas Instruments TI59 calculator, and the evaluation
of $L'(E,1)$ was done on a CDC Cyber computer at the University of Virginia.}

\section{A Rank~2 Example}
According to \cite{Ma-SwD} the first rank~2 curve with prime conductor
is
\[
E\qquad y^2+y=x^3+x^2-2x, \qquad N=\Delta=389.
\]
By reducing modulo 2, 3 and 5 it is easy to check that $E_{\rm tor}(\Q)$
is zero. (This also follows from a lemma in \cite{Ma-SwD} since
$E(\Q)$ contains more than 6 points.) Let $A=(0,0)$ and $B=(-1,1)$
be rational points in $E(\Q)$. It is easy to check that $A$ and $B$ are
both distinct and non-zero in the finite quotient $E(\Q)/2E(\Q)$.
On the other hand, the cubic subfield $F$ of $\Q(E[2])$ has discriminant
$4\cdot389$ and ring of integers $\Z[\theta]$ where 
$\theta^3-\theta^2-9\theta+11=0$. Using the Minkowski bound and the 
factorizations of 2, 3, 5 and 7 in $\Z[\theta]$ one sees that $F$ has
class number~1. The rank bounds of \cite{Br-Kr} then imply
that $\rank E(\Q)\leq 2$. Thus we have $\rank E(\Q)=2$, and
the bounds just referred to also show that $\Sha^{(2)}(E/\Q)=0$ also.

Since we have $c_{389}=1$, the conjecture here says that firstly
$\Ls$ has an analytic continuation,  has a zero of order~2 at $s=1$, and secondly
that
\[
L''(E,1) =2\Omega\left|\Sha\right|\det\left(\begin{array}{cc} \hgt{A} & \hgtp{A}{B}\\
\hgtp{B}{A} & \hgt{B} \end{array}\right).
\]
(We are assuming here that $A$ and $B$ actually generate $E(\Q)$, which could
be readily checked.)
The height pairing $\hgtp{A}{B}=\hgtp{B}{A}=h^{*}(A+B)-h^{*}(A)-h^{*}(B)$
and the height and period computations give:
\begin{eqnarray*}
\Omega & = & 4.980425122,  \\
\hgt{A} = 2h^{*}(A) & = & 2\lambda_{\infty}(A) = .3270007736, \\
\hgt{B} = 2h^{*}(B) & = & 2\lambda_{\infty}(B) = .6866670834, \\
\hgtp{A}{B} =\hgtp{B}{A} &= & \lambda_{\infty}(A+B)-\lambda_{\infty}(A)
-\lambda_{\infty}(B) \\
         & = & -.2684780988,
\end{eqnarray*}
and we find that the period times the height determinant is 1.518633001.

We assume for the purposes of the computation that $E$ is indeed a Weil curve. Since
it has prime conductor, and the tangents to the node in $\tilde{E}(\F{389})$ are
$\F{389}$ rational, the sign in the functional equation must be $+1$, and the order
of vanishing of the $L$-series is even.  Using the integral representation for
$\Phi(s)=(2\pi)^{-s}\Gamma(s)\Ls$ given by
\[
\Phi(s) = \int_{p^{-1/2}}^{\infty} (t^{s-1}+p^{1-s}t^{1-s})f(it)\,dt,
\]
where $f$ is the newform corresponding to $E$, $f(z)=\sum_n a_nq^n$, we find
that
\[
L(E,1) = 2\sum_{n=1}^{\infty} \frac{a_n}{n}\, e^{-2\pi n/\sqrt{389}}.
\]
Furthermore,  if we know that $L(E,1)=0$, then integrating the formula for $\Phi(s)$
by parts  after differentiating twice gives
\[
L''(E,1) =4\sum_{n=1}^{\infty} \frac{a_n}{n} 
\int_{1/\sqrt{389}}^{\infty}\log(\sqrt{389}t)\, e^{-2\pi nt}\frac{dt}{t}.
\]
It is then routine to find the $a_n$, and then to evaluate these series. Using 130 terms
gives $|L(E,1)| <10^{-8}$ and that $L''(E,1)=1.518633009$, agreeing with the 
period times height term.  And the predicted $\Sha$ is trivial here.

\appendix
\section{Computing Periods of Elliptic Curves}
We describe how to compute the fundamental period of an elliptic curve using Gauss'
arithmetic geometric mean.
 % incorporate paper on periods here, slightly condensed
\section{Computing Heights}
We recall the definition and main properties of heights of points on an elliptic curve,
and give a method of Tate's to calculate the local height function at the infinite real
place of $\Q$.
 % brief description of heights, refer to Silverman for more
\section{Computing $L$-series}
% adapt old notes
This summarizes some estimates needed to evaluate the $L$-series derivative values
accurately.

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