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\title{Math 168A: MIDTERM}
\author{William Stein}
\date{\bf Due: Wednesday, Oct 26, 2005}
\begin{document}
\maketitle

\noindent{\em The problems have equal point value, and parts of multi-part
  problems are of the same value.
{\bf You may not talk to anybody about these problems.}  You are allowed
to use your notes, computer software, books and web pages.
}

\begin{enumerate}
\item Make a conjecture about the set of primes $p$
such that $y^2 = x^3 + x$ has exactly $p+1$ points
modulo $p$.
You do not have to prove your conjecture.  
You might find the following table helpful:
%
% E = EllipticCurve([1,0])
% t = [(p, E.Np(p)) for p in primes(100)]
% sage: ' & '.join([str(a) for a, _ in t])
% _7 = '2 & 3 & 5 & 7 & 11 & 13 & 17 & 19 & 23 & 29 & 31 & 37 & 41 & 43 & 47 & 53 & 59 & 61 & 67 & 71 & 73 & 79 & 83 & 89 & 97'
%sage: ' & '.join([str(b) for _, b in t])
% _9 = '3 & 4 & 4 & 8 & 12 & 20 & 16 & 20 & 24 & 20 & 32 & 36 & 32 & 44 & 48 & 68 & 60 & 52 & 68 & 72 & 80 & 80 & 84 & 80 & 80'
%sage: '|'.join(['c' for _, b in t])
%_10 = 'c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c'
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline
$p$ & 2 & 3 & 5 & 7 & 11 & 13 & 17 & 19 & 23 & 29 & 31 & 37 & 41 \\\hline
$N_p$ & 3 & 4 & 4 & 8 & 12 & 20 & 16 & 20 & 24 & 20 & 32 & 36 & 32 \\\hline
\end{tabular}

\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline
$p$ & 43 & 47 & 53 & 59 & 61 & 67 & 71 & 73 & 79 & 83 & 89 & 97 \\\hline
$N_p$ &44 & 48 & 68 & 60 & 52 & 68 & 72 & 80 & 80 & 84 & 80 & 80\\\hline
\end{tabular}
\end{center}

\item The point $(1,1)$ is a solution to the equation
\begin{equation}\label{eqn:0}
  3 + 2y + y^2 = x^2 + 2x + 3x^3.
\end{equation}
Find three other solutions as follows:
\begin{enumerate}
\item Multiply both sides of the equation by $9$
and let $X=3x$, $Y=3y$ to obtain an equation
of the form 
\begin{equation}\label{eqn:1}
  Y^2 + a_1 XY + a_3 Y = X^3 + a_2X^2 + a_4 X + a_6.
\end{equation}
\item Find the point $P$ on (\ref{eqn:1}) corresponding
to the point $(1,1)$ on (\ref{eqn:0}).
\item Use the group operation on (\ref{eqn:1}), e.g., via
SAGE, to compute several multiples of $P$.

\item Use the multiples of $P$ to recover points on (\ref{eqn:0}).

\end{enumerate}


\item The right triangle with rational side 
lengths $a=24/5$, $b=35/12$, $c=337/60$ has area $7$.
Find another right triangle with rational side lengths
and area $7$ as follows (note that none of the sides
lengths of this second triangle are allowed to be the same
as the side lengths of the above triangle):

\begin{enumerate}
\item  Find the point $P$ on $y^2 = x^3 - 49x$ corresponding
to the triangle $(24/5, 35/12, 337/60)$ under the bijection
from class (see notes from 2005-10-17).

\item Compute $Q = P+P$ using the elliptic curve group law.

\item Find the triangle corresponding to $Q$ under
the bijection from class.

\end{enumerate}

\end{enumerate}
\end{document}
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