% 3/19/90, 3/20/90
%
% slide of equations and conductors
%
\message{(Don't forget to magnify right!)}
\magnification\magstep1
\global\overfullrule0pt
\def\endpage{\vfill\eject}
\nopagenumbers
\vsize = 4.0 truein
\hsize = 7.5 truein
\hoffset = -0.2 truein
\voffset = 1.0 truein
%
\def\term#1#2{\ifnum#1<0 \ifnum#1=-1 -#2 \else #1 #2 \fi\fi
													\ifnum#1=0  {} \fi
													\ifnum#1=1  + #2  \fi
													\ifnum#1>1  + #1 #2\fi}
\def\constterm#1{\ifnum#1<0 #1 \fi
                \ifnum#1=0 {} \fi
																\ifnum#1>0 +#1\fi}
%
\def\curve[#1,#2,#3,#4,#5]{$y^2\term{#1}{xy}\term{#3}{y}
				=x^3\term{#2}{x^2}\term{#4}{x}\constterm{#5}$}
%

\centerline{\bf CURVES OF SMALLEST KNOWN CONDUCTORS FOR EACH RANK}
\vskip 0.3in
$$\vbox{\offinterlineskip
\hrule
\halign{&\vrule#&\strut\quad\hfil\bf#\hfil\quad\cr
height2pt&\omit&&\omit&&\omit&&\omit&\cr
&Rank &&Conductor&& Equation && Generators &\cr 
height2pt&\omit&&\omit&&\omit&&\omit&\cr
\noalign{\hrule}
height2pt&\omit&&\omit&&\omit&&\omit&\cr
&0 && 11 && \curve[0,-1,1,0,0]&&0&\cr
&1 && 37 && \curve[0,0,1,-1,0]&&0&\cr
&2 && 389 && \curve[0,1,1, -2,0]&&0, $-$1&\cr
&3 && 5077 && \curve[0,0,1,-7,6]&&1, 2, 0&\cr
&4 && 501029 &&\curve[0,1,1,-72,210]&&5, 4, 3, 6&\cr
&5 && 19047851 &&\curve[0,0,1,-79,342]&&5, 4, 3, 7, 0&\cr
&6 && 6756532597 && \curve[0,0,1,-547, -2934]&&$-$6, $-$7, $-$8, $-$9, $-$13,
$-$14&\cr 
height2pt&\omit&&\omit&&\omit&&\omit&\cr}
\hrule}
$$
\vskip0.2in
%\centerline{\bf For the rank~0 curve, a generator of the torsion group is given.}
%\centerline{\bf The generators are given in order of increasing height.}
%\centerline{\bf The rank 0, 5 and 6 curves have negative discriminants.}
\medskip
%\centerline{\bf The rank 4, rank 5 and rank 6 curves are believed to be new.}
\message{(Don't forget to magnify right!)}
\bye