Sharedwww / tables / serremodpq / serremodpq.texOpen in CoCalc
Author: William A. Stein
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% serremodpq.tex
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\documentclass{article}
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\title{Serre's Conjecture Mod $pq$?}
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\author{B. Mazur\and W.A. Stein}
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\date{January 11, 1999}
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\newcommand{\comment}[1]{}
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\begin{document}
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\maketitle
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\tableofcontents
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\section{Introduction}
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I haven't yet written an introduction, but here are some helpful remarks
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as to how to read the table. Please see Barry Mazur's letter below
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which motivated the computation. Basically, when both an $f$ and $g$
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occur together in a given row, then ``Serre's Conjecture mod $pq$''
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for a certain representation has an easy positive answer.
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The rows of the congruence table contain an $f$ whenever the
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corresponding eigenform
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at level $NM$ is congruent to the reduction of $f$ modulo the prime $P$ listed
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in the first column, at least to the precision computed.
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Similarly for $g$. In some cases the program did not perform
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a complete check that the forms were congruent to the precision
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computed, but only checked, with {\em high} probability,
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that there is a congruence to that precision.
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In such cases this is indicated
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by a question mark subscript.
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In many cases Ribet's theorem implies
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that there must be a congruence.
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I do not know if Sturm's result can be applied in
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this context because the congruence is only checked
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for those $a_p$ for which $p$ is prime to $NMpq$.
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In a few cases it was too much extra work to reduce the prime level $\leq 97$
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eigenforms modulo $2$, and so this is not done, even though $2$ might
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be among the canidate pairs $(P, Q)$. This is indicated above the
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relevant table.
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The eigenforms refered to in the congruence table are uniquely
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determined, up to conjugacy, by a factor of the characteristic
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polynomial of some Hecke operator. These factors are given, so that
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further investigation of particular eigenforms is possible.
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To understand why this table was computed, see the letter from Barry
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Mazur in the last section. Note that the congruence table does {\em not}
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omit primes for which the corresponding representations are reducible.
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\section{Letters}
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\newcommand{\Q}{\mathbf{Q}}
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\newcommand{\GL}{{\rm GL}}
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\newcommand{\Z}{\mathbf{Z}}
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\newcommand{\mod}{{\rm\,\, mod\,\,}}
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\begin{verbatim}
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From: Barry Mazur
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Subject: SERRE'S CONJECTURE MOD PQ
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To: William Stein
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\end{verbatim}
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Dear William,
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To put perspective on the subject of this e-mail, let me begin with
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an utterly untestable version of what I was suggesting to you. Let $p$
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and $q$ be distinct prime numbers and $G$ the Galois group of the maximal
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algebraic extension of $\Q$ unramified outside a finite set $S$ of primes.
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Suppose that we have an odd continuous representation
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$r: G \rightarrow \GL_2(\Z/pq\Z)$
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which is irreducible when reduced mod $p$ and mod $q$. Is $r$ "modular"?
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To begin to give {\em some} shape to the above question, we might as well
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assume the
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``classical'' Serre conjecture for single primes, so we can start out by
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giving
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ourselves two modular newforms $f$ and $g$ and then look for a modular newform
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$h$ that
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is congruent mod $p$ to $f$ and mod $q$ to $g$ -- under the assumption that the
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associated
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Galois representations mod $p$ and mod $q$ respectively are irreducible. I
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should
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be more careful about what I mean ``mod $p$'' and ``mod $q$'': I really mean to
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take $p$ and $q$ prime ideals in the ring generated by the Fourier coefficients
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of $f$ and $g$, and ditto with regard to $h$. To simplify one's life, one
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can "start out" by working only with pairs $f$ and $g$ of newforms whose ring
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of Fourier coefficients is $\Z$ (i.e., the $f$ and $g$ in question correspond to
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elliptic curves). But even if one does this, one has to face the
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possibility
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(the ``probability'', in fact) that $h$, if it exists at all, no longer has its
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ring of
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Fourier coefficients equal to $\Z$.
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Of course, phrased this way, our qeustion is still relatively
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un``falsifiable''
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so we have better be more specific.
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Ken and I thought a bit about how to shape a more specific testable
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question,
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and here is what we came up with-- very tentatively still. Suppose that $N$
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and $M$ are
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distinct prime numbers, and $f$ and $g$ are newforms of weight two on
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$X_0(N)$
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and $X_0(M)$
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respectively. Now let $p$ be a prime number (or ideal) dividing either
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$1+M- a_M(f)$ or
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$1+M + a_M(f)$, and let $q$ be a prime number (or ideal) dividing either
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$1+N - a_N(g)$ or $1+N + a_N(g)$. Let us suppose that the residual
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characteristics
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of the ideals $p$ and $q$ are distinct (i.e., if we are in a case where $p$ and
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$q$ are
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prime numbers, we just want them to be distinct). Suppose also that the
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corresponding Galois representations associated to $f \mod p$ and $g \mod q$
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are irreducible. Let us say that we have ``maximum success''
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if for any such choice of $f,g, p, q$ there is a modular newform $h$ of weight
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two on $X_0(NM)$
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which is congruent to $f \mod p$ and to $g \mod q$.
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Here $a_N(f)$ means the $N$th Fourier coefficient of $f$, etc.
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The reason why we call the above ``maximum success'' is that we are
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putting down
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the ``evident'' conditions necessary for there to be such a newform $h$ (and
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looking
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for such an $h$ at the minimal conceivable level, and weight). It is
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probably
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unreasonable to expect ``maximum success'', but initial experiments might
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give us a sense of
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how far from the mark ``maximal success'' actually is, in which case we will
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weaken our
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expectations accordingly... I should also say, that the first case is
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$N=11$ and $M=17$,
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where the level of the sought for $h$ is $187$, i.e., not too large yet...
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Does this sound as if it is a conceivable, fun, etc. computation to
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make?
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Barry
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\begin{verbatim}
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Date: Mon, 26 Oct 1998 12:46:09 -0500
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\end{verbatim}
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Dear William,
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That's terrific! Actually I had debated about whether to exclude the
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prime $2$ from the search-- it being slightly problematic for level-raising,
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level-lowering issues. I also excluded reducible representations as being
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significantly more problematic on a number of other grounds. But, while you
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are systemtically doing the computation, it might make sense not to
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exclude reducible representations from the search, but just indicate
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in the data when they are reducible. After all, we may also be getting
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some
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``maximal succeses'' there too (but if we get a "failure" for a
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reducible case, I wouldn't worry). Keep me posted!
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Barry
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\begin{verbatim}
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Date: Tue, 3 Nov 1998 20:32:35 +0000 (GMT)
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From: Kevin Buzzard <buzzard@ic.ac.uk>
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To: was@bmw.autobahn.org
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Subject: Re: congruence table
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I looked through your tables a fair bit yesterday and can't find anything
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which smacks of contradiction, which is very comforting. It all seems
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to fit together very well, which means that I am getting more and more
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confident in your programs!
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[NB my experience with these programs is that if they "usually" work
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then they still might not work 100% because if numbers get too big
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then there's weird overflows and so maybe things don't work then---
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but I don't have any evidence at all that your programs don't
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work 100%]
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Anyway, here's some random comments about them. When you say
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in the intro that "in _some_ cases the program didn't perform
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a complete check of congruence"---really, you never performed
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a _complete_ check. If you mean "complete check up to p=***"
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whatever *** is, maybe you should say this. Also it's a bit
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strange that you then go on to talk about the ?? cases, and
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say that they're congruences with a "high probability", whereas
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in reality we're *always* only dealing with probabilities (except,
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I guess, in cases where Ken's theorem predicts that a form
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exists and only 1 form can possibly work at level NM). In practice
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it's very difficult to verify that forms are actually congruent
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because the effective Chebatorev bounds aren't good enough.
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\end{verbatim}
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\newpage
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\textheight=1.28\textheight
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\textwidth=1.35\textwidth
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\voffset=-1.3in
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\hoffset=-.9in
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\section{Table of Congruences}
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\noindent\begin{minipage}{\textwidth}
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\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=17A}$}\\
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\begin{tabular}{|cc|cccccc|}\hline
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$P$&$Q$&187A&187B&187C&187D&187E&187F\\\hline
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$2$&$3$&$ f$&$ f$&$ fg$&$ f$&$ f$&$ g$\\\hline
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$5$&$2$&$ $&$ $&$ $&$ $&$ g$&$ $\\\hline
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$5$&$3$&$ $&$ $&$ g$&$ $&$ $&$ g$\\\hline
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\end{tabular}\end{minipage}
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\noindent\begin{minipage}{\textwidth}
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\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=19A}$}\\
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\begin{tabular}{|cc|cccc|}\hline
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$P$&$Q$&209A&209B&209C&209D\\\hline
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$2$&$3$&$ f$&$ f_?$&$ f_?g$&$ fg$\\\hline
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$2$&$5$&$ f$&$ f_?$&$ f_?g$&$ f$\\\hline
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$5$&$3$&$ $&$ f$&$ g$&$ g$\\\hline
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\end{tabular}\end{minipage}
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\noindent\begin{minipage}{\textwidth}
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\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=23A}$}\\
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\begin{tabular}{|cc|cccc|}\hline
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$P$&$Q$&253A&253B&253C&253D\\\hline
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$23$&$({x + {8}},11)
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$&$ $&$ $&$ $&$ f$\\\hline
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$23$&$({x + {15}},19)
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$&$ g$&$ $&$ $&$ f$\\\hline
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$23$&$({x^{2} + x + 1},2)
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$&$ $&$ $&$ g$&$ fg$\\\hline
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$23$&$({x + {3}},5)
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$&$ $&$ g$&$ $&$ f$\\\hline
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$5$&$({x + {8}},11)
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$&$ $&$ f$&$ $&$ $\\\hline
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$5$&$({x + {15}},19)
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$&$ g$&$ f$&$ $&$ $\\\hline
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$5$&$({x^{2} + x + 1},2)
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$&$ $&$ f$&$ g$&$ g$\\\hline
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\end{tabular}\end{minipage}
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\noindent\begin{minipage}{\textwidth}
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\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=29A}$}\\
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\begin{tabular}{|cc|ccccc|}\hline
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$P$&$Q$&319A&319B&319C&319D&319E\\\hline
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$2$&$({x + {14}},167)
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$&$ f$&$ $&$ f$&$ $&$ g$\\\hline
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$2$&$({x + {7}},17)
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$&$ f$&$ g$&$ f$&$ $&$ $\\\hline
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$2$&$({x + {4}},7)
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$&$ f$&$ $&$ f$&$ $&$ $\\\hline
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$3$&$({x + {14}},167)
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$&$ $&$ f$&$ $&$ f$&$ g$\\\hline
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$3$&$({x + {7}},17)
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$&$ $&$ fg$&$ $&$ f$&$ $\\\hline
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$3$&$({x + {4}},7)
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$&$ $&$ f$&$ $&$ f$&$ $\\\hline
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$5$&$({x + {14}},167)
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$&$ $&$ $&$ $&$ f$&$ g$\\\hline
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$5$&$({x + {7}},17)
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$&$ $&$ g$&$ $&$ f$&$ $\\\hline
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$5$&$({x + {4}},7)
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$&$ $&$ $&$ $&$ f$&$ $\\\hline
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\end{tabular}\end{minipage}
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\noindent\begin{minipage}{\textwidth}
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\vspace{.2in}\noindent{\large $\mathbf{f=17A, \, g=19A}$}\\
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\begin{tabular}{|cc|cccccc|}\hline
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$P$&$Q$&323A&323B&323C&323D&323E&323F\\\hline
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$2$&$3$&$ f$&$ $&$ f$&$ f_?g$&$ f_?$&$ $\\\hline
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$2$&$5$&$ f$&$ $&$ f$&$ f_?$&$ f_?$&$ g$\\\hline
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$2$&$7$&$ f$&$ g$&$ f$&$ f_?$&$ f_?$&$ $\\\hline
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$3$&$5$&$ $&$ f$&$ $&$ $&$ $&$ g$\\\hline
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$3$&$7$&$ $&$ fg$&$ $&$ $&$ $&$ $\\\hline
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\end{tabular}\end{minipage}
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\noindent\begin{minipage}{\textwidth}
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\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=31A}$}\\
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\begin{tabular}{|cc|cccc|}\hline
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$P$&$Q$&341A&341B&341C&341D\\\hline
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$13$&$({x^{2} + x + 1},2)
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$&$ $&$ g_?$&$ $&$ fg_?$\\\hline
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$13$&$({x + {2}},5)
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$&$ g$&$ $&$ $&$ f$\\\hline
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$13$&$({x^{2} + {6}x + {6}},7)
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$&$ $&$ $&$ g_?$&$ f$\\\hline
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$3$&$({x^{2} + x + 1},2)
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$&$ $&$ g_?$&$ $&$ f_?g_?$\\\hline
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$3$&$({x + {2}},5)
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$&$ g$&$ $&$ $&$ f_?$\\\hline
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$3$&$({x^{2} + {6}x + {6}},7)
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$&$ $&$ $&$ g_?$&$ f_?$\\\hline
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$5$&$({x^{2} + x + 1},2)
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$&$ f$&$ g_?$&$ $&$ g_?$\\\hline
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$5$&$({x^{2} + {6}x + {6}},7)
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$&$ f$&$ $&$ g_?$&$ $\\\hline
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\end{tabular}\end{minipage}
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\noindent\begin{minipage}{\textwidth}
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\vspace{.2in}\noindent{\large $\mathbf{f=17A, \, g=23A}$}\\
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\begin{tabular}{|cc|ccccc|}\hline
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$P$&$Q$&391A&391B&391C&391D&391E\\\hline
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$2$&$({x + {99}},109)
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$&$ $&$ $&$ g$&$ $&$ f_?$\\\hline
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$2$&$({x + {8}},11)
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$&$ $&$ $&$ $&$ $&$ f_?$\\\hline
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$2$&$({x + {3}},5)
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$&$ $&$ g$&$ $&$ $&$ f_?$\\\hline
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$5$&$({x + {99}},109)
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$&$ $&$ f$&$ g$&$ $&$ $\\\hline
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$5$&$({x + {8}},11)
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$&$ $&$ f$&$ $&$ $&$ $\\\hline
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$5$&$({x^{2} + x + 1},2)
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$&$ $&$ f$&$ $&$ g$&$ g_?$\\\hline
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$7$&$({x + {99}},109)
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$&$ $&$ $&$ g$&$ $&$ f_?$\\\hline
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$7$&$({x + {8}},11)
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$&$ $&$ $&$ $&$ $&$ f_?$\\\hline
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$7$&$({x^{2} + x + 1},2)
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$&$ $&$ $&$ $&$ g$&$ f_?g_?$\\\hline
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$7$&$({x + {3}},5)
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$&$ $&$ g$&$ $&$ $&$ f_?$\\\hline
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\end{tabular}\end{minipage}
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\noindent\begin{minipage}{\textwidth}
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\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=37A}$}\\
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\begin{tabular}{|cc|cccc|}\hline
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$P$&$Q$&407A&407B&407C&407D\\\hline
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$41$&$17$&$ $&$ $&$ fg$&$ $\\\hline
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$41$&$7$&$ g$&$ $&$ f$&$ $\\\hline
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$5$&$17$&$ $&$ $&$ g$&$ $\\\hline
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$5$&$7$&$ g$&$ $&$ $&$ $\\\hline
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$7$&$17$&$ $&$ f$&$ g$&$ $\\\hline
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\end{tabular}\end{minipage}
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\noindent\begin{minipage}{\textwidth}
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\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=37B}$}\\
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\begin{tabular}{|cc|cccc|}\hline
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$P$&$Q$&407A&407B&407C&407D\\\hline
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$41$&$3$&$ $&$ g$&$ f$&$ g$\\\hline
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$41$&$5$&$ $&$ $&$ f$&$ g$\\\hline
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$5$&$3$&$ $&$ g$&$ $&$ g$\\\hline
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$7$&$3$&$ $&$ fg$&$ $&$ g$\\\hline
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$7$&$5$&$ $&$ f$&$ $&$ g$\\\hline
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\end{tabular}\end{minipage}
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\noindent\begin{minipage}{\textwidth}
353
\vspace{.2in}\noindent{\large $\mathbf{f=19A, \, g=23A}$}\\
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\begin{tabular}{|cc|cccccccc|}\hline
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$P$&$Q$&437A&437B&437C&437D&437E&437F&437G&437H\\\hline
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$2$&$({x + {4}},11)
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$&$ f$&$ f$&$ g$&$ f$&$ $&$ f_?$&$ f_?$&$ f_?$\\\hline
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$2$&$({x + {8}},11)
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$&$ f$&$ f$&$ $&$ f$&$ $&$ f_?$&$ f_?$&$ f_?$\\\hline
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$2$&$({x^{2} + x + {2}},3)
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$&$ f$&$ f$&$ $&$ f$&$ $&$ f_?$&$ f_?g$&$ f_?$\\\hline
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$3$&$({x + {4}},11)
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$&$ $&$ $&$ g$&$ $&$ $&$ $&$ $&$ f$\\\hline
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$3$&$({x + {8}},11)
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$&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ f$\\\hline
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$3$&$({x^{2} + x + 1},2)
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$&$ $&$ $&$ $&$ $&$ $&$ g_?$&$ $&$ fg_?$\\\hline
368
\end{tabular}\end{minipage}
369
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371
372
\noindent\begin{minipage}{\textwidth}
373
\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=41A}$} {\small (char 2 primes for $g$ omitted)}\\
374
\begin{tabular}{|cc|ccccc|}\hline
375
$P$&$Q$&451A&451B&451C&451D&451E\\\hline
376
$17$&$({x + {2}},5)
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$&$ $&$ g$&$ f$&$ $&$ $\\\hline
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$17$&$({x + {413}},863)
379
$&$ $&$ $&$ f$&$ g$&$ $\\\hline
380
$2$&$({x + {2}},5)
381
$&$ $&$ g$&$ $&$ f_?$&$ f$\\\hline
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$2$&$({x + {413}},863)
383
$&$ $&$ $&$ $&$ f_?g$&$ f$\\\hline
384
$5$&$({x + {413}},863)
385
$&$ $&$ f$&$ $&$ g$&$ $\\\hline
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\end{tabular}\end{minipage}
387
388
389
390
\noindent\begin{minipage}{\textwidth}
391
\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=43A}$}\\
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\begin{tabular}{|cc|ccccccc|}\hline
393
$P$&$Q$&473A&473B&473C&473D&473E&473F&473G\\\hline
394
$19$&$3$&$ $&$ $&$ $&$ $&$ $&$ fg$&$ g$\\\hline
395
$19$&$5$&$ $&$ g$&$ $&$ $&$ $&$ f$&$ $\\\hline
396
$2$&$3$&$ $&$ $&$ $&$ $&$ f_?$&$ g$&$ fg$\\\hline
397
$2$&$5$&$ $&$ g$&$ $&$ $&$ f_?$&$ $&$ f$\\\hline
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$5$&$3$&$ f$&$ $&$ $&$ $&$ $&$ g$&$ g$\\\hline
399
\end{tabular}\end{minipage}
400
401
\noindent\begin{minipage}{\textwidth}
402
\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=43B}$}\\
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\begin{tabular}{|cc|ccccccc|}\hline
404
$P$&$Q$&473A&473B&473C&473D&473E&473F&473G\\\hline
405
$19$&$({x + {62}},113)
406
$&$ $&$ $&$ $&$ $&$ g$&$ f$&$ $\\\hline
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$19$&$({x + {5}},23)
408
$&$ $&$ $&$ g$&$ $&$ $&$ f$&$ $\\\hline
409
$19$&$({x + {4}},7)
410
$&$ $&$ $&$ $&$ $&$ $&$ f$&$ $\\\hline
411
$2$&$({x + {62}},113)
412
$&$ $&$ $&$ $&$ $&$ f_?g$&$ $&$ f$\\\hline
413
$2$&$({x + {5}},23)
414
$&$ $&$ $&$ g$&$ $&$ f_?$&$ $&$ f$\\\hline
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$2$&$({x + {4}},7)
416
$&$ $&$ $&$ $&$ $&$ f_?$&$ $&$ f$\\\hline
417
$5$&$({x + {62}},113)
418
$&$ f$&$ $&$ $&$ $&$ g$&$ $&$ $\\\hline
419
$5$&$({x + {5}},23)
420
$&$ f$&$ $&$ g$&$ $&$ $&$ $&$ $\\\hline
421
$5$&$({x + {4}},7)
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$&$ f$&$ $&$ $&$ $&$ $&$ $&$ $\\\hline
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\end{tabular}\end{minipage}
424
425
426
427
\noindent\begin{minipage}{\textwidth}
428
\vspace{.2in}\noindent{\large $\mathbf{f=17A, \, g=29A}$}\\
429
\begin{tabular}{|cc|cccccccc|}\hline
430
$P$&$Q$&493A&493B&493C&493D&493E&493F&493G&493H\\\hline
431
$2$&$({x + {24}},31)
432
$&$ $&$ $&$ $&$ $&$ f$&$ f$&$ f_?g$&$ $\\\hline
433
$2$&$({x + {4}},7)
434
$&$ $&$ $&$ $&$ g$&$ f$&$ f$&$ f_?$&$ $\\\hline
435
$3$&$({x + 1},2)
436
$&$ fg$&$ g$&$ g$&$ $&$ $&$ g$&$ $&$ f$\\\hline
437
$3$&$({x + {24}},31)
438
$&$ f$&$ $&$ $&$ $&$ $&$ $&$ g$&$ f$\\\hline
439
$3$&$({x + {4}},7)
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$&$ f$&$ $&$ $&$ g$&$ $&$ $&$ $&$ f$\\\hline
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\end{tabular}\end{minipage}
442
443
444
445
\noindent\begin{minipage}{\textwidth}
446
\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=47A}$}\\
447
\begin{tabular}{|cc|ccccccccccc|}\hline
448
$P$&$Q$&517A&517B&517C&517D&517E&517F&517G&517H&517I&517J&517K\\\hline
449
$2$&$({x + {20}},23)
450
$&$ f$&$ f$&$ f_?$&$ $&$ $&$ f_?$&$ $&$ $&$ f_?$&$ f_?$&$ f$\\\hline
451
$2$&$({x + 1},3)
452
$&$ f$&$ fg$&$ f_?$&$ $&$ $&$ f_?g$&$ $&$ g_?$&$ f_?$&$ f_?$&$ fg$\\\hline
453
$2$&$({x + {8}},31)
454
$&$ f$&$ f$&$ f_?g$&$ $&$ $&$ f_?$&$ $&$ $&$ f_?$&$ f_?$&$ f$\\\hline
455
$2$&$({x + {2}},7)
456
$&$ f$&$ f$&$ f_?$&$ g$&$ $&$ f_?$&$ $&$ $&$ f_?$&$ f_?$&$ f$\\\hline
457
$5$&$({x^{4} + x^{3} + x^{2} + x + 1},2)
458
$&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ g_?$&$ g_?$&$ $\\\hline
459
$5$&$({x + {20}},23)
460
$&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $\\\hline
461
$5$&$({x + 1},3)
462
$&$ $&$ g$&$ $&$ $&$ $&$ g$&$ $&$ g_?$&$ $&$ $&$ g$\\\hline
463
$5$&$({x + {8}},31)
464
$&$ $&$ $&$ g$&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $\\\hline
465
$5$&$({x + {2}},7)
466
$&$ $&$ $&$ $&$ g$&$ $&$ $&$ $&$ $&$ $&$ $&$ $\\\hline
467
$7$&$({x^{4} + x^{3} + x^{2} + x + 1},2)
468
$&$ $&$ $&$ $&$ $&$ f$&$ $&$ $&$ $&$ g_?$&$ g_?$&$ $\\\hline
469
$7$&$({x + {20}},23)
470
$&$ $&$ $&$ $&$ $&$ f$&$ $&$ $&$ $&$ $&$ $&$ $\\\hline
471
$7$&$({x + 1},3)
472
$&$ $&$ g$&$ $&$ $&$ f$&$ g$&$ $&$ g_?$&$ $&$ $&$ g$\\\hline
473
$7$&$({x + {8}},31)
474
$&$ $&$ $&$ g$&$ $&$ f$&$ $&$ $&$ $&$ $&$ $&$ $\\\hline
475
\end{tabular}\end{minipage}
476
477
478
479
\noindent\begin{minipage}{\textwidth}
480
\vspace{.2in}\noindent{\large $\mathbf{f=17A, \, g=31A}$}\\
481
\begin{tabular}{|cc|ccccc|}\hline
482
$P$&$Q$&527A&527B&527C&527D&527E\\\hline
483
$2$&$({x + {98}},109)
484
$&$ $&$ $&$ $&$ g$&$ f_?$\\\hline
485
$2$&$({x + {7}},11)
486
$&$ g$&$ $&$ $&$ $&$ f_?$\\\hline
487
$2$&$({x + {2}},5)
488
$&$ $&$ $&$ $&$ $&$ f_?$\\\hline
489
$3$&$({x + {98}},109)
490
$&$ $&$ $&$ $&$ g$&$ f_?$\\\hline
491
$3$&$({x + {7}},11)
492
$&$ g$&$ $&$ $&$ $&$ f_?$\\\hline
493
$3$&$({x^{2} + x + 1},2)
494
$&$ $&$ $&$ g$&$ $&$ f_?g_?$\\\hline
495
$3$&$({x + {2}},5)
496
$&$ $&$ $&$ $&$ $&$ f_?$\\\hline
497
$7$&$({x + {98}},109)
498
$&$ f$&$ $&$ $&$ g$&$ $\\\hline
499
$7$&$({x + {7}},11)
500
$&$ fg$&$ $&$ $&$ $&$ $\\\hline
501
$7$&$({x^{2} + x + 1},2)
502
$&$ f$&$ $&$ g$&$ $&$ g_?$\\\hline
503
$7$&$({x + {2}},5)
504
$&$ f$&$ $&$ $&$ $&$ $\\\hline
505
\end{tabular}\end{minipage}
506
507
508
509
510
\noindent\begin{minipage}{\textwidth}
511
\vspace{.2in}\noindent{\large $\mathbf{f=19A, \, g=29A}$}\\
512
\begin{tabular}{|cc|cccccccc|}\hline
513
$P$&$Q$&551A&551B&551C&551D&551E&551F&551G&551H\\\hline
514
$2$&$({x^{2} + {2}x + {12}},13)
515
$&$ $&$ f$&$ f$&$ $&$ f$&$ f_?$&$ f_?g$&$ $\\\hline
516
$2$&$({x + {4}},7)
517
$&$ $&$ f$&$ f$&$ $&$ f$&$ f_?$&$ f_?$&$ $\\\hline
518
$2$&$({x + {5}},7)
519
$&$ $&$ f$&$ fg$&$ $&$ f$&$ f_?$&$ f_?$&$ $\\\hline
520
$3$&$({x^{2} + {2}x + {12}},13)
521
$&$ $&$ $&$ $&$ $&$ $&$ f_?$&$ g$&$ $\\\hline
522
$3$&$({x + 1},2)
523
$&$ g$&$ $&$ $&$ g$&$ $&$ f_?g_?$&$ g_?$&$ g$\\\hline
524
$3$&$({x + {4}},7)
525
$&$ $&$ $&$ $&$ $&$ $&$ f_?$&$ $&$ $\\\hline
526
$3$&$({x + {5}},7)
527
$&$ $&$ $&$ g$&$ $&$ $&$ f_?$&$ $&$ $\\\hline
528
\end{tabular}\end{minipage}
529
530
531
532
\noindent\begin{minipage}{\textwidth}
533
\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=53A}$}\\
534
\begin{tabular}{|cc|ccccccccc|}\hline
535
$P$&$Q$&583A&583B&583C&583D&583E&583F&583G&583H&583I\\\hline
536
$2$&$3$&$ $&$ f$&$ fg$&$ f$&$ fg$&$ $&$ f_?$&$ f_?$&$ f_?$\\\hline
537
$3$&$2$&$ fg$&$ $&$ $&$ $&$ $&$ g_?$&$ f$&$ g_?$&$ $\\\hline
538
$5$&$2$&$ g$&$ $&$ $&$ $&$ $&$ g_?$&$ $&$ g_?$&$ $\\\hline
539
$5$&$3$&$ $&$ $&$ g$&$ $&$ g$&$ $&$ $&$ $&$ $\\\hline
540
\end{tabular}\end{minipage}
541
542
\noindent\begin{minipage}{\textwidth}
543
\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=53B}$}\\
544
\begin{tabular}{|cc|ccccccccc|}\hline
545
$P$&$Q$&583A&583B&583C&583D&583E&583F&583G&583H&583I\\\hline
546
$2$&$({x + {10}},13)
547
$&$ $&$ f$&$ f$&$ f$&$ f$&$ $&$ f_?$&$ f_?$&$ f_?$\\\hline
548
$2$&$({x + {224}},281)
549
$&$ $&$ f$&$ f$&$ f$&$ f$&$ $&$ f_?$&$ f_?$&$ f_?g$\\\hline
550
$2$&$({x + {5}},43)
551
$&$ $&$ f$&$ f$&$ f$&$ f$&$ $&$ f_?g$&$ f_?$&$ f_?$\\\hline
552
$3$&$({x + {10}},13)
553
$&$ f$&$ $&$ $&$ $&$ $&$ $&$ f$&$ $&$ $\\\hline
554
$3$&$({x + 1},2)
555
$&$ fg$&$ $&$ $&$ $&$ $&$ g_?$&$ f$&$ g_?$&$ $\\\hline
556
$3$&$({x + {224}},281)
557
$&$ f$&$ $&$ $&$ $&$ $&$ $&$ f$&$ $&$ g$\\\hline
558
$3$&$({x + {5}},43)
559
$&$ f$&$ $&$ $&$ $&$ $&$ $&$ fg$&$ $&$ $\\\hline
560
$5$&$({x + {10}},13)
561
$&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $\\\hline
562
$5$&$({x + 1},2)
563
$&$ g$&$ $&$ $&$ $&$ $&$ g_?$&$ $&$ g_?$&$ $\\\hline
564
$5$&$({x + {224}},281)
565
$&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ $&$ g$\\\hline
566
$5$&$({x + {5}},43)
567
$&$ $&$ $&$ $&$ $&$ $&$ $&$ g$&$ $&$ $\\\hline
568
\end{tabular}\end{minipage}
569
570
571
572
\noindent\begin{minipage}{\textwidth}
573
\vspace{.2in}\noindent{\large $\mathbf{f=19A, \, g=31A}$}\\
574
\begin{tabular}{|cc|ccccc|}\hline
575
$P$&$Q$&589A&589B&589C&589D&589E\\\hline
576
$2$&$({x + {2}},5)
577
$&$ $&$ f$&$ f$&$ f_?g$&$ f$\\\hline
578
$2$&$({x + {29}},79)
579
$&$ $&$ f$&$ f$&$ f_?g$&$ f$\\\hline
580
$2$&$({x + {49}},79)
581
$&$ $&$ f$&$ f$&$ f_?$&$ fg$\\\hline
582
$3$&$({x + {2}},5)
583
$&$ $&$ $&$ $&$ g$&$ f$\\\hline
584
$3$&$({x + {29}},79)
585
$&$ $&$ $&$ $&$ g$&$ f$\\\hline
586
$3$&$({x + {49}},79)
587
$&$ $&$ $&$ $&$ $&$ fg$\\\hline
588
$7$&$({x + {2}},5)
589
$&$ f$&$ $&$ $&$ g$&$ $\\\hline
590
$7$&$({x + {29}},79)
591
$&$ f$&$ $&$ $&$ g$&$ $\\\hline
592
$7$&$({x + {49}},79)
593
$&$ f$&$ $&$ $&$ $&$ g$\\\hline
594
\end{tabular}\end{minipage}
595
596
597
598
\noindent\begin{minipage}{\textwidth}
599
\vspace{.2in}\noindent{\large $\mathbf{f=17A, \, g=37A}$}\\
600
\begin{tabular}{|cc|cccccccc|}\hline
601
$P$&$Q$&629A&629B&629C&629D&629E&629F&629G&629H\\\hline
602
$2$&$3$&$ g$&$ $&$ $&$ $&$ f_?$&$ f_?g$&$ fg$&$ $\\\hline
603
$3$&$2$&$ $&$ $&$ fg$&$ $&$ g_?$&$ f$&$ g$&$ g$\\\hline
604
$5$&$2$&$ $&$ f$&$ g$&$ $&$ g_?$&$ $&$ g$&$ g$\\\hline
605
$5$&$3$&$ g$&$ f$&$ $&$ $&$ $&$ g$&$ g$&$ $\\\hline
606
\end{tabular}\end{minipage}
607
608
\noindent\begin{minipage}{\textwidth}
609
\vspace{.2in}\noindent{\large $\mathbf{f=17A, \, g=37B}$}\\
610
\begin{tabular}{|cc|cccccccc|}\hline
611
$P$&$Q$&629A&629B&629C&629D&629E&629F&629G&629H\\\hline
612
$2$&$3$&$ $&$ $&$ $&$ $&$ f_?g$&$ f_?$&$ f$&$ $\\\hline
613
$3$&$2$&$ $&$ $&$ fg$&$ $&$ g_?$&$ f$&$ g$&$ g$\\\hline
614
$5$&$2$&$ $&$ f$&$ g$&$ $&$ g_?$&$ $&$ g$&$ g$\\\hline
615
$5$&$3$&$ $&$ f$&$ $&$ $&$ g$&$ $&$ $&$ $\\\hline
616
\end{tabular}\end{minipage}
617
618
619
620
\noindent\begin{minipage}{\textwidth}
621
\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=59A}$} {\small (char 2 primes for $g$ omitted)}\\
622
\begin{tabular}{|cc|cccccc|}\hline
623
$P$&$Q$&649A&649B&649C&649D&649E&649F\\\hline
624
$11$&$({x + {9}},19)
625
$&$ $&$ $&$ fg$&$ $&$ $&$ $\\\hline
626
$11$&$({x + {1940}},2671)
627
$&$ $&$ $&$ f$&$ g$&$ $&$ $\\\hline
628
$11$&$({x + {26}},29)
629
$&$ $&$ $&$ f$&$ $&$ $&$ $\\\hline
630
$11$&$({x + 1},7)
631
$&$ $&$ g$&$ f$&$ $&$ $&$ $\\\hline
632
$13$&$({x + {9}},19)
633
$&$ $&$ $&$ g$&$ $&$ f$&$ $\\\hline
634
$13$&$({x + {1940}},2671)
635
$&$ $&$ $&$ $&$ g$&$ f$&$ $\\\hline
636
$13$&$({x + {26}},29)
637
$&$ $&$ $&$ $&$ $&$ f$&$ $\\\hline
638
$13$&$({x + 1},7)
639
$&$ $&$ g$&$ $&$ $&$ f$&$ $\\\hline
640
$5$&$({x + {9}},19)
641
$&$ $&$ $&$ g$&$ $&$ f$&$ $\\\hline
642
$5$&$({x + {1940}},2671)
643
$&$ $&$ $&$ $&$ g$&$ f$&$ $\\\hline
644
$5$&$({x + {26}},29)
645
$&$ $&$ $&$ $&$ $&$ f$&$ $\\\hline
646
$5$&$({x + 1},7)
647
$&$ $&$ g$&$ $&$ $&$ f$&$ $\\\hline
648
\end{tabular}\end{minipage}
649
650
651
652
\noindent\begin{minipage}{\textwidth}
653
\vspace{.2in}\noindent{\large $\mathbf{f=23A, \, g=29A}$}\\
654
\begin{tabular}{|cc|cccc|}\hline
655
$P$&$Q$&667A&667B&667C&667D\\\hline
656
$({x + {4}},11)
657
$&$({x + {52}},113)
658
$&$ f$&$ $&$ $&$ g$\\\hline
659
$({x + {4}},11)
660
$&$({x + 1},2)
661
$&$ f$&$ g$&$ g_?$&$ $\\\hline
662
$({x + {4}},11)
663
$&$({x + {19}},23)
664
$&$ fg$&$ $&$ $&$ $\\\hline
665
$({x + {4}},11)
666
$&$({x + {4}},7)
667
$&$ f$&$ $&$ $&$ $\\\hline
668
$({x + {8}},11)
669
$&$({x + {52}},113)
670
$&$ $&$ $&$ $&$ g$\\\hline
671
$({x + {8}},11)
672
$&$({x + 1},2)
673
$&$ $&$ g$&$ g_?$&$ $\\\hline
674
$({x + {8}},11)
675
$&$({x + {19}},23)
676
$&$ g$&$ $&$ $&$ $\\\hline
677
$({x + {8}},11)
678
$&$({x + {4}},7)
679
$&$ $&$ $&$ $&$ $\\\hline
680
$({x^{2} + x + {2}},3)
681
$&$({x + {52}},113)
682
$&$ f$&$ $&$ f$&$ g$\\\hline
683
$({x^{2} + x + {2}},3)
684
$&$({x + 1},2)
685
$&$ f$&$ g$&$ fg_?$&$ $\\\hline
686
$({x^{2} + x + {2}},3)
687
$&$({x + {19}},23)
688
$&$ fg$&$ $&$ f$&$ $\\\hline
689
$({x^{2} + x + {2}},3)
690
$&$({x + {4}},7)
691
$&$ f$&$ $&$ f$&$ $\\\hline
692
\end{tabular}\end{minipage}
693
694
695
696
\noindent\begin{minipage}{\textwidth}
697
\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=61A}$}\\
698
\begin{tabular}{|cc|cccc|}\hline
699
$P$&$Q$&671A&671B&671C&671D\\\hline
700
$2$&$17$&$ $&$ f$&$ f$&$ g$\\\hline
701
$2$&$7$&$ $&$ fg$&$ f$&$ $\\\hline
702
$37$&$17$&$ $&$ $&$ $&$ fg$\\\hline
703
$37$&$7$&$ $&$ g$&$ $&$ f$\\\hline
704
$5$&$17$&$ f$&$ $&$ $&$ g$\\\hline
705
$5$&$7$&$ f$&$ g$&$ $&$ $\\\hline
706
\end{tabular}\end{minipage}
707
708
\noindent\begin{minipage}{\textwidth}
709
\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=61B}$}\\
710
\begin{tabular}{|cc|cccc|}\hline
711
$P$&$Q$&671A&671B&671C&671D\\\hline
712
$2$&$({x + {3}},13)
713
$&$ $&$ f$&$ fg$&$ $\\\hline
714
$2$&$({x + {9}},17)
715
$&$ g$&$ f$&$ f$&$ $\\\hline
716
$2$&$({x + {16}},331)
717
$&$ $&$ f$&$ fg$&$ $\\\hline
718
$2$&$({x + {2}},5)
719
$&$ g$&$ f$&$ f$&$ $\\\hline
720
$37$&$({x + {3}},13)
721
$&$ $&$ $&$ g$&$ f$\\\hline
722
$37$&$({x + {9}},17)
723
$&$ g$&$ $&$ $&$ f$\\\hline
724
$37$&$({x + {16}},331)
725
$&$ $&$ $&$ g$&$ f$\\\hline
726
$37$&$({x + {2}},5)
727
$&$ g$&$ $&$ $&$ f$\\\hline
728
$5$&$({x + {3}},13)
729
$&$ f$&$ $&$ g$&$ $\\\hline
730
$5$&$({x + {9}},17)
731
$&$ fg$&$ $&$ $&$ $\\\hline
732
$5$&$({x + {16}},331)
733
$&$ f$&$ $&$ g$&$ $\\\hline
734
\end{tabular}\end{minipage}
735
736
737
738
\noindent\begin{minipage}{\textwidth}
739
\vspace{.2in}\noindent{\large $\mathbf{f=17A, \, g=41A}$} {\small (char 2 primes for $g$ omitted)}\\
740
\begin{tabular}{|cc|cccccc|}\hline
741
$P$&$Q$&697A&697B&697C&697D&697E&697F\\\hline
742
$2$&$({x + {2}},5)
743
$&$ f$&$ f_?$&$ f$&$ $&$ f_?$&$ f_?$\\\hline
744
$2$&$({x^{2} + {4}x + {2}},5)
745
$&$ f$&$ f_?$&$ f$&$ g$&$ f_?$&$ f_?$\\\hline
746
$3$&$({x + {2}},5)
747
$&$ f$&$ $&$ $&$ $&$ f$&$ $\\\hline
748
$3$&$({x^{2} + {4}x + {2}},5)
749
$&$ f$&$ $&$ $&$ g$&$ f$&$ $\\\hline
750
\end{tabular}\end{minipage}
751
752
753
754
\noindent\begin{minipage}{\textwidth}
755
\vspace{.2in}\noindent{\large $\mathbf{f=19A, \, g=37A}$}\\
756
\begin{tabular}{|cc|ccccccccc|}\hline
757
$P$&$Q$&703A&703B&703C&703D&703E&703F&703G&703H&703I\\\hline
758
$2$&$5$&$ f$&$ $&$ $&$ $&$ $&$ f_?$&$ fg$&$ f_?g$&$ $\\\hline
759
$3$&$2$&$ $&$ g$&$ g$&$ f$&$ g$&$ g_?$&$ $&$ g_?$&$ g$\\\hline
760
$3$&$5$&$ $&$ $&$ $&$ f$&$ $&$ $&$ g$&$ g$&$ $\\\hline
761
$5$&$2$&$ $&$ g$&$ g$&$ $&$ g$&$ g_?$&$ $&$ g_?$&$ fg$\\\hline
762
\end{tabular}\end{minipage}
763
764
\noindent\begin{minipage}{\textwidth}
765
\vspace{.2in}\noindent{\large $\mathbf{f=19A, \, g=37B}$}\\
766
\begin{tabular}{|cc|ccccccccc|}\hline
767
$P$&$Q$&703A&703B&703C&703D&703E&703F&703G&703H&703I\\\hline
768
$2$&$11$&$ f$&$ $&$ $&$ $&$ $&$ f_?g$&$ f$&$ f_?$&$ $\\\hline
769
$2$&$3$&$ f$&$ $&$ $&$ g$&$ $&$ f_?$&$ f$&$ f_?$&$ $\\\hline
770
$3$&$11$&$ $&$ $&$ $&$ f$&$ $&$ g$&$ $&$ $&$ $\\\hline
771
$3$&$2$&$ $&$ g$&$ g$&$ f$&$ g$&$ g_?$&$ $&$ g_?$&$ g$\\\hline
772
$5$&$11$&$ $&$ $&$ $&$ $&$ $&$ g$&$ $&$ $&$ f$\\\hline
773
$5$&$2$&$ $&$ g$&$ g$&$ $&$ g$&$ g_?$&$ $&$ g_?$&$ fg$\\\hline
774
$5$&$3$&$ $&$ $&$ $&$ g$&$ $&$ $&$ $&$ $&$ f$\\\hline
775
\end{tabular}\end{minipage}
776
777
778
779
\noindent\begin{minipage}{\textwidth}
780
\vspace{.2in}\noindent{\large $\mathbf{f=23A, \, g=31A}$}\\
781
\begin{tabular}{|cc|ccccccc|}\hline
782
$P$&$Q$&713A&713B&713C&713D&713E&713F&713G\\\hline
783
$({x + {4}},11)
784
$&$({x^{2} + x + 1},2)
785
$&$ $&$ g$&$ $&$ $&$ fg_?$&$ $&$ $\\\hline
786
$({x + {4}},11)
787
$&$({x + {5}},29)
788
$&$ $&$ $&$ $&$ $&$ f$&$ $&$ g$\\\hline
789
$({x + {4}},11)
790
$&$({x + {2}},5)
791
$&$ $&$ $&$ $&$ $&$ f$&$ $&$ $\\\hline
792
$({x + {8}},11)
793
$&$({x^{2} + x + 1},2)
794
$&$ $&$ g$&$ $&$ $&$ g_?$&$ $&$ $\\\hline
795
$({x + {8}},11)
796
$&$({x + {5}},29)
797
$&$ $&$ $&$ $&$ $&$ $&$ $&$ g$\\\hline
798
$({x + {8}},11)
799
$&$({x + {2}},5)
800
$&$ $&$ $&$ $&$ $&$ $&$ $&$ $\\\hline
801
$({x + {10}},89)
802
$&$({x + {7}},11)
803
$&$ $&$ $&$ $&$ g$&$ $&$ $&$ f$\\\hline
804
$({x + {10}},89)
805
$&$({x^{2} + x + 1},2)
806
$&$ $&$ g$&$ $&$ $&$ g_?$&$ $&$ f$\\\hline
807
$({x + {10}},89)
808
$&$({x + {5}},29)
809
$&$ $&$ $&$ $&$ $&$ $&$ $&$ fg$\\\hline
810
$({x + {10}},89)
811
$&$({x + {2}},5)
812
$&$ $&$ $&$ $&$ $&$ $&$ $&$ f$\\\hline
813
$({x + {80}},89)
814
$&$({x + {7}},11)
815
$&$ $&$ $&$ $&$ g$&$ f_?$&$ $&$ $\\\hline
816
$({x + {80}},89)
817
$&$({x^{2} + x + 1},2)
818
$&$ $&$ g$&$ $&$ $&$ f_?g_?$&$ $&$ $\\\hline
819
$({x + {80}},89)
820
$&$({x + {5}},29)
821
$&$ $&$ $&$ $&$ $&$ f_?$&$ $&$ g$\\\hline
822
$({x + {80}},89)
823
$&$({x + {2}},5)
824
$&$ $&$ $&$ $&$ $&$ f_?$&$ $&$ $\\\hline
825
\end{tabular}\end{minipage}
826
827
828
829
830
831
\noindent\begin{minipage}{\textwidth}
832
\vspace{.2in}\noindent{\large $\mathbf{f=17A, \, g=43A}$}\\
833
\begin{tabular}{|cc|cccccc|}\hline
834
$P$&$Q$&731A&731B&731C&731D&731E&731F\\\hline
835
$2$&$3$&$ g$&$ f$&$ $&$ $&$ f_?$&$ f_?g_?$\\\hline
836
$2$&$5$&$ $&$ f$&$ g$&$ $&$ f_?$&$ f_?$\\\hline
837
$2$&$7$&$ $&$ f$&$ $&$ $&$ f_?$&$ f_?g$\\\hline
838
$3$&$5$&$ $&$ $&$ g$&$ $&$ $&$ f_?$\\\hline
839
$3$&$7$&$ $&$ $&$ $&$ $&$ $&$ f_?g$\\\hline
840
$5$&$3$&$ g$&$ $&$ $&$ f$&$ $&$ g_?$\\\hline
841
$5$&$7$&$ $&$ $&$ $&$ f$&$ $&$ g$\\\hline
842
\end{tabular}\end{minipage}
843
844
\noindent\begin{minipage}{\textwidth}
845
\vspace{.2in}\noindent{\large $\mathbf{f=17A, \, g=43B}$}\\
846
\begin{tabular}{|cc|cccccc|}\hline
847
$P$&$Q$&731A&731B&731C&731D&731E&731F\\\hline
848
$2$&$({x + {5}},23)
849
$&$ $&$ f$&$ $&$ g$&$ f_?$&$ f_?$\\\hline
850
$2$&$({x + {272}},521)
851
$&$ $&$ f$&$ $&$ $&$ f_?g$&$ f_?$\\\hline
852
$2$&$({x + {4}},7)
853
$&$ $&$ f$&$ $&$ $&$ f_?$&$ f_?$\\\hline
854
$3$&$({x + {5}},23)
855
$&$ $&$ $&$ $&$ g$&$ $&$ f_?$\\\hline
856
$3$&$({x + {272}},521)
857
$&$ $&$ $&$ $&$ $&$ g$&$ f_?$\\\hline
858
$3$&$({x + {4}},7)
859
$&$ $&$ $&$ $&$ $&$ $&$ f_?$\\\hline
860
$5$&$({x + {5}},23)
861
$&$ $&$ $&$ $&$ fg$&$ $&$ $\\\hline
862
$5$&$({x + {272}},521)
863
$&$ $&$ $&$ $&$ f$&$ g$&$ $\\\hline
864
$5$&$({x + {4}},7)
865
$&$ $&$ $&$ $&$ f$&$ $&$ $\\\hline
866
\end{tabular}\end{minipage}
867
868
869
870
\noindent\begin{minipage}{\textwidth}
871
\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=67A}$}\\
872
\begin{tabular}{|cc|cccccc|}\hline
873
$P$&$Q$&737A&737B&737C&737D&737E&737F\\\hline
874
$3$&$2$&$ fg$&$ g$&$ g_?$&$ g$&$ g_?$&$ $\\\hline
875
$5$&$2$&$ g$&$ g$&$ g_?$&$ g$&$ g_?$&$ f$\\\hline
876
$61$&$2$&$ g$&$ g$&$ fg_?$&$ g$&$ g_?$&$ $\\\hline
877
\end{tabular}\end{minipage}
878
879
\noindent\begin{minipage}{\textwidth}
880
\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=67B}$}\\
881
\begin{tabular}{|cc|cccccc|}\hline
882
$P$&$Q$&737A&737B&737C&737D&737E&737F\\\hline
883
$3$&$({x + {4}},11)
884
$&$ f$&$ $&$ $&$ $&$ $&$ g$\\\hline
885
$3$&$({x + {8}},11)
886
$&$ f$&$ $&$ $&$ $&$ $&$ $\\\hline
887
$3$&$({x^{2} + x + {12}},13)
888
$&$ f$&$ $&$ $&$ $&$ g$&$ $\\\hline
889
$5$&$({x + {4}},11)
890
$&$ $&$ $&$ $&$ $&$ $&$ fg$\\\hline
891
$5$&$({x + {8}},11)
892
$&$ $&$ $&$ $&$ $&$ $&$ f$\\\hline
893
$5$&$({x^{2} + x + {12}},13)
894
$&$ $&$ $&$ $&$ $&$ g$&$ f$\\\hline
895
$61$&$({x + {4}},11)
896
$&$ $&$ $&$ f$&$ $&$ $&$ g$\\\hline
897
$61$&$({x + {8}},11)
898
$&$ $&$ $&$ f$&$ $&$ $&$ $\\\hline
899
$61$&$({x^{2} + x + {12}},13)
900
$&$ $&$ $&$ f$&$ $&$ g$&$ $\\\hline
901
\end{tabular}\end{minipage}
902
903
\noindent\begin{minipage}{\textwidth}
904
\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=67C}$}\\
905
\begin{tabular}{|cc|cccccc|}\hline
906
$P$&$Q$&737A&737B&737C&737D&737E&737F\\\hline
907
$3$&$({x + {65}},139)
908
$&$ f$&$ $&$ $&$ g$&$ $&$ $\\\hline
909
$3$&$({x + {77}},139)
910
$&$ f$&$ $&$ g$&$ $&$ $&$ $\\\hline
911
$5$&$({x + {65}},139)
912
$&$ $&$ $&$ $&$ g$&$ $&$ f$\\\hline
913
$5$&$({x + {77}},139)
914
$&$ $&$ $&$ g$&$ $&$ $&$ f$\\\hline
915
$61$&$({x + {65}},139)
916
$&$ $&$ $&$ f$&$ g$&$ $&$ $\\\hline
917
$61$&$({x + {77}},139)
918
$&$ $&$ $&$ fg$&$ $&$ $&$ $\\\hline
919
\end{tabular}\end{minipage}
920
921
922
923
\noindent\begin{minipage}{\textwidth}
924
\vspace{.2in}\noindent{\large $\mathbf{f=19A, \, g=41A}$} {\small (char 2 primes for $g$ omitted)}\\
925
\begin{tabular}{|cc|ccccc|}\hline
926
$P$&$Q$&779A&779B&779C&779D&779E\\\hline
927
$2$&$({x + {2}},5)
928
$&$ f$&$ f$&$ f$&$ f_?$&$ f_?g_?$\\\hline
929
$2$&$({x + {171}},607)
930
$&$ f$&$ fg$&$ f$&$ f_?$&$ f_?$\\\hline
931
$2$&$({x + {299}},929)
932
$&$ f$&$ f$&$ f$&$ f_?$&$ f_?g$\\\hline
933
$3$&$({x + {2}},5)
934
$&$ $&$ $&$ $&$ f$&$ g_?$\\\hline
935
$3$&$({x + {171}},607)
936
$&$ $&$ g$&$ $&$ f$&$ $\\\hline
937
$3$&$({x + {299}},929)
938
$&$ $&$ $&$ $&$ f$&$ g$\\\hline
939
\end{tabular}\end{minipage}
940
941
942
943
\noindent\begin{minipage}{\textwidth}
944
\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=71A}$}\\
945
\begin{tabular}{|cc|ccccccc|}\hline
946
$P$&$Q$&781A&781B&781C&781D&781E&781F&781G\\\hline
947
$23$&$({x^{3} + x + 1},2)
948
$&$ $&$ $&$ $&$ $&$ g_?$&$ $&$ fg_?$\\\hline
949
$23$&$({x},3)
950
$&$ g$&$ g$&$ g$&$ g$&$ $&$ $&$ f$\\\hline
951
$23$&$({x + {2}},5)
952
$&$ $&$ $&$ $&$ g$&$ $&$ $&$ f$\\\hline
953
$23$&$({x + {20}},53)
954
$&$ $&$ $&$ $&$ $&$ $&$ g$&$ f$\\\hline
955
$3$&$({x^{3} + x + 1},2)
956
$&$ $&$ $&$ f$&$ $&$ g_?$&$ $&$ f_?g_?$\\\hline
957
$3$&$({x + {2}},5)
958
$&$ $&$ $&$ f$&$ g$&$ $&$ $&$ f_?$\\\hline
959
$3$&$({x + {20}},53)
960
$&$ $&$ $&$ f$&$ $&$ $&$ g$&$ f_?$\\\hline
961
$5$&$({x^{3} + x + 1},2)
962
$&$ $&$ $&$ $&$ f$&$ g_?$&$ $&$ g_?$\\\hline
963
$5$&$({x},3)
964
$&$ g$&$ g$&$ g$&$ fg$&$ $&$ $&$ $\\\hline
965
$5$&$({x + {20}},53)
966
$&$ $&$ $&$ $&$ f$&$ $&$ g$&$ $\\\hline
967
\end{tabular}\end{minipage}
968
969
\noindent\begin{minipage}{\textwidth}
970
\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=71B}$}\\
971
\begin{tabular}{|cc|ccccccc|}\hline
972
$P$&$Q$&781A&781B&781C&781D&781E&781F&781G\\\hline
973
$23$&$({x^{3} + x^{2} + 1},2)
974
$&$ $&$ $&$ $&$ $&$ g_?$&$ $&$ fg_?$\\\hline
975
$23$&$({x},3)
976
$&$ g$&$ g$&$ g$&$ g$&$ $&$ $&$ f$\\\hline
977
$23$&$({x + {27}},61)
978
$&$ $&$ $&$ $&$ $&$ $&$ g_?$&$ f$\\\hline
979
$23$&$({x + {4}},7)
980
$&$ $&$ $&$ $&$ $&$ $&$ $&$ f$\\\hline
981
$3$&$({x^{3} + x^{2} + 1},2)
982
$&$ $&$ $&$ f$&$ $&$ g_?$&$ $&$ f_?g_?$\\\hline
983
$3$&$({x + {27}},61)
984
$&$ $&$ $&$ f$&$ $&$ $&$ g_?$&$ f_?$\\\hline
985
$3$&$({x + {4}},7)
986
$&$ $&$ $&$ f$&$ $&$ $&$ $&$ f_?$\\\hline
987
$5$&$({x^{3} + x^{2} + 1},2)
988
$&$ $&$ $&$ $&$ f$&$ g_?$&$ $&$ g_?$\\\hline
989
$5$&$({x},3)
990
$&$ g$&$ g$&$ g$&$ fg$&$ $&$ $&$ $\\\hline
991
$5$&$({x + {27}},61)
992
$&$ $&$ $&$ $&$ f$&$ $&$ g_?$&$ $\\\hline
993
$5$&$({x + {4}},7)
994
$&$ $&$ $&$ $&$ f$&$ $&$ $&$ $\\\hline
995
\end{tabular}\end{minipage}
996
997
998
\noindent\begin{minipage}{\textwidth}
999
\vspace{.2in}\noindent{\large $\mathbf{f=17A, \, g=47A}$}\\
1000
\begin{tabular}{|cc|ccccccc|}\hline
1001
$P$&$Q$&799A&799B&799C&799D&799E&799F&799G\\\hline
1002
$2$&$({x + {20}},23)
1003
$&$ f$&$ $&$ f$&$ f_?$&$ $&$ f_?$&$ f$\\\hline
1004
$2$&$({x + 1},3)
1005
$&$ f$&$ $&$ f$&$ f_?g_?$&$ $&$ f_?$&$ fg$\\\hline
1006
$2$&$({x + {3495}},43991)
1007
$&$ f$&$ $&$ f$&$ f_?g$&$ $&$ f_?$&$ f$\\\hline
1008
$2$&$({x + {55}},937)
1009
$&$ f$&$ $&$ f$&$ f_?$&$ g$&$ f_?$&$ f$\\\hline
1010
$3$&$({x + {20}},23)
1011
$&$ $&$ $&$ $&$ $&$ f$&$ f$&$ $\\\hline
1012
$3$&$({x + {3495}},43991)
1013
$&$ $&$ $&$ $&$ g$&$ f$&$ f$&$ $\\\hline
1014
$3$&$({x + {55}},937)
1015
$&$ $&$ $&$ $&$ $&$ fg$&$ f$&$ $\\\hline
1016
\end{tabular}\end{minipage}
1017
1018
1019
1020
\noindent\begin{minipage}{\textwidth}
1021
\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=73A}$}\\
1022
\begin{tabular}{|cc|cccccc|}\hline
1023
$P$&$Q$&803A&803B&803C&803D&803E&803F\\\hline
1024
$13$&$2$&$ $&$ $&$ f$&$ g_?$&$ $&$ $\\\hline
1025
$13$&$5$&$ $&$ $&$ f$&$ g$&$ $&$ $\\\hline
1026
$13$&$7$&$ $&$ $&$ f$&$ $&$ $&$ g$\\\hline
1027
$2$&$5$&$ $&$ f$&$ $&$ f_?g$&$ f$&$ $\\\hline
1028
$2$&$7$&$ $&$ f$&$ $&$ $&$ f$&$ g$\\\hline
1029
$3$&$2$&$ $&$ $&$ f_?$&$ g_?$&$ $&$ $\\\hline
1030
$3$&$5$&$ $&$ $&$ f_?$&$ g$&$ $&$ $\\\hline
1031
$3$&$7$&$ $&$ $&$ f_?$&$ $&$ $&$ g$\\\hline
1032
$5$&$2$&$ $&$ $&$ $&$ g_?$&$ $&$ $\\\hline
1033
$5$&$7$&$ $&$ $&$ $&$ $&$ $&$ g$\\\hline
1034
$7$&$2$&$ $&$ $&$ $&$ g_?$&$ $&$ f$\\\hline
1035
$7$&$5$&$ $&$ $&$ $&$ g$&$ $&$ f$\\\hline
1036
\end{tabular}\end{minipage}
1037
1038
\noindent\begin{minipage}{\textwidth}
1039
\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=73B}$}\\
1040
\begin{tabular}{|cc|cccccc|}\hline
1041
$P$&$Q$&803A&803B&803C&803D&803E&803F\\\hline
1042
$13$&$({x + {14}},23)
1043
$&$ $&$ $&$ f$&$ $&$ $&$ g$\\\hline
1044
$13$&$({x},3)
1045
$&$ $&$ g$&$ f$&$ $&$ $&$ $\\\hline
1046
$13$&$({x + {15}},79)
1047
$&$ $&$ $&$ f$&$ g$&$ $&$ $\\\hline
1048
$2$&$({x + {14}},23)
1049
$&$ $&$ f$&$ $&$ $&$ f$&$ g$\\\hline
1050
$2$&$({x},3)
1051
$&$ $&$ fg$&$ $&$ $&$ f$&$ $\\\hline
1052
$2$&$({x + {15}},79)
1053
$&$ $&$ f$&$ $&$ g$&$ f$&$ $\\\hline
1054
$3$&$({x + {14}},23)
1055
$&$ $&$ $&$ f_?$&$ $&$ $&$ g$\\\hline
1056
$3$&$({x + {15}},79)
1057
$&$ $&$ $&$ f_?$&$ g$&$ $&$ $\\\hline
1058
$5$&$({x + {14}},23)
1059
$&$ $&$ $&$ $&$ $&$ $&$ g$\\\hline
1060
$5$&$({x},3)
1061
$&$ $&$ g$&$ $&$ $&$ $&$ $\\\hline
1062
$5$&$({x + {15}},79)
1063
$&$ $&$ $&$ $&$ g$&$ $&$ $\\\hline
1064
$7$&$({x + {14}},23)
1065
$&$ $&$ $&$ $&$ $&$ $&$ fg$\\\hline
1066
$7$&$({x},3)
1067
$&$ $&$ g$&$ $&$ $&$ $&$ f$\\\hline
1068
$7$&$({x + {15}},79)
1069
$&$ $&$ $&$ $&$ g$&$ $&$ f$\\\hline
1070
\end{tabular}\end{minipage}
1071
1072
\noindent\begin{minipage}{\textwidth}
1073
\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=73C}$}\\
1074
\begin{tabular}{|cc|cccccc|}\hline
1075
$P$&$Q$&803A&803B&803C&803D&803E&803F\\\hline
1076
$13$&$({x + {100}},109)
1077
$&$ $&$ $&$ f$&$ $&$ g$&$ $\\\hline
1078
$13$&$({x + {15}},181)
1079
$&$ $&$ $&$ fg$&$ $&$ $&$ $\\\hline
1080
$2$&$({x + {100}},109)
1081
$&$ $&$ f$&$ $&$ $&$ fg$&$ $\\\hline
1082
$2$&$({x + {15}},181)
1083
$&$ $&$ f$&$ g$&$ $&$ f$&$ $\\\hline
1084
$3$&$({x + {100}},109)
1085
$&$ $&$ $&$ f_?$&$ $&$ g$&$ $\\\hline
1086
$3$&$({x + {15}},181)
1087
$&$ $&$ $&$ f_?g$&$ $&$ $&$ $\\\hline
1088
$5$&$({x + {100}},109)
1089
$&$ $&$ $&$ $&$ $&$ g$&$ $\\\hline
1090
$5$&$({x + {15}},181)
1091
$&$ $&$ $&$ g$&$ $&$ $&$ $\\\hline
1092
$7$&$({x + {100}},109)
1093
$&$ $&$ $&$ $&$ $&$ g$&$ f$\\\hline
1094
$7$&$({x + {15}},181)
1095
$&$ $&$ $&$ g$&$ $&$ $&$ f$\\\hline
1096
\end{tabular}\end{minipage}
1097
1098
1099
1100
\noindent\begin{minipage}{\textwidth}
1101
\vspace{.2in}\noindent{\large $\mathbf{f=19A, \, g=43A}$}\\
1102
\begin{tabular}{|cc|cccccc|}\hline
1103
$P$&$Q$&817A&817B&817C&817D&817E&817F\\\hline
1104
$3$&$11$&$ $&$ $&$ f_?$&$ $&$ $&$ g_?$\\\hline
1105
$3$&$2$&$ g$&$ g$&$ f_?$&$ g_?$&$ $&$ g_?$\\\hline
1106
$43$&$11$&$ $&$ $&$ $&$ $&$ $&$ fg_?$\\\hline
1107
$43$&$2$&$ g$&$ g$&$ $&$ g_?$&$ $&$ fg_?$\\\hline
1108
$43$&$3$&$ $&$ $&$ $&$ g$&$ $&$ f$\\\hline
1109
$5$&$11$&$ f$&$ $&$ $&$ $&$ $&$ g_?$\\\hline
1110
$5$&$2$&$ fg$&$ g$&$ $&$ g_?$&$ $&$ g_?$\\\hline
1111
$5$&$3$&$ f$&$ $&$ $&$ g$&$ $&$ $\\\hline
1112
\end{tabular}\end{minipage}
1113
1114
\noindent\begin{minipage}{\textwidth}
1115
\vspace{.2in}\noindent{\large $\mathbf{f=19A, \, g=43B}$}\\
1116
\begin{tabular}{|cc|cccccc|}\hline
1117
$P$&$Q$&817A&817B&817C&817D&817E&817F\\\hline
1118
$3$&$({x + {11}},17)
1119
$&$ $&$ $&$ f_?g_?$&$ $&$ $&$ $\\\hline
1120
$3$&$({x},2)
1121
$&$ g$&$ g$&$ f_?$&$ g_?$&$ $&$ g_?$\\\hline
1122
$3$&$({x + {4}},7)
1123
$&$ $&$ $&$ f_?$&$ $&$ $&$ $\\\hline
1124
$3$&$({x + {70}},79)
1125
$&$ $&$ $&$ f_?$&$ $&$ g$&$ $\\\hline
1126
$43$&$({x + {11}},17)
1127
$&$ $&$ $&$ g_?$&$ $&$ $&$ f$\\\hline
1128
$43$&$({x},2)
1129
$&$ g$&$ g$&$ $&$ g_?$&$ $&$ fg_?$\\\hline
1130
$43$&$({x + {4}},7)
1131
$&$ $&$ $&$ $&$ $&$ $&$ f$\\\hline
1132
$43$&$({x + {70}},79)
1133
$&$ $&$ $&$ $&$ $&$ g$&$ f$\\\hline
1134
$5$&$({x + {11}},17)
1135
$&$ f$&$ $&$ g_?$&$ $&$ $&$ $\\\hline
1136
$5$&$({x},2)
1137
$&$ fg$&$ g$&$ $&$ g_?$&$ $&$ g_?$\\\hline
1138
$5$&$({x + {4}},7)
1139
$&$ f$&$ $&$ $&$ $&$ $&$ $\\\hline
1140
$5$&$({x + {70}},79)
1141
$&$ f$&$ $&$ $&$ $&$ g$&$ $\\\hline
1142
\end{tabular}\end{minipage}
1143
1144
1145
1146
\noindent\begin{minipage}{\textwidth}
1147
\vspace{.2in}\noindent{\large $\mathbf{f=23A, \, g=37A}$}\\
1148
\begin{tabular}{|cc|cccccccc|}\hline
1149
$P$&$Q$&851A&851B&851C&851D&851E&851F&851G&851H\\\hline
1150
$({x + {8}},11)
1151
$&$13$&$ $&$ $&$ $&$ $&$ $&$ g$&$ $&$ $\\\hline
1152
$({x + {8}},11)
1153
$&$2$&$ g$&$ $&$ $&$ g$&$ g_?$&$ $&$ $&$ g$\\\hline
1154
$({x^{2} + x + 1},2)
1155
$&$11$&$ $&$ $&$ $&$ $&$ g$&$ f_?$&$ f_?$&$ $\\\hline
1156
$({x^{2} + x + 1},2)
1157
$&$13$&$ $&$ $&$ $&$ $&$ $&$ f_?g$&$ f_?$&$ $\\\hline
1158
$({x + {19}},31)
1159
$&$11$&$ $&$ $&$ $&$ f$&$ g$&$ $&$ $&$ $\\\hline
1160
$({x + {19}},31)
1161
$&$13$&$ $&$ $&$ $&$ f$&$ $&$ g$&$ $&$ $\\\hline
1162
$({x + {19}},31)
1163
$&$2$&$ g$&$ $&$ $&$ fg$&$ g_?$&$ $&$ $&$ g$\\\hline
1164
$({x + {360}},379)
1165
$&$11$&$ $&$ $&$ $&$ $&$ fg$&$ $&$ $&$ $\\\hline
1166
$({x + {360}},379)
1167
$&$13$&$ $&$ $&$ $&$ $&$ f$&$ g$&$ $&$ $\\\hline
1168
$({x + {360}},379)
1169
$&$2$&$ g$&$ $&$ $&$ g$&$ fg_?$&$ $&$ $&$ g$\\\hline
1170
\end{tabular}\end{minipage}
1171
1172
\noindent\begin{minipage}{\textwidth}
1173
\vspace{.2in}\noindent{\large $\mathbf{f=23A, \, g=37B}$}\\
1174
\begin{tabular}{|cc|cccccccc|}\hline
1175
$P$&$Q$&851A&851B&851C&851D&851E&851F&851G&851H\\\hline
1176
$({x + {8}},11)
1177
$&$2$&$ g$&$ $&$ $&$ g$&$ g_?$&$ $&$ $&$ g$\\\hline
1178
$({x + {8}},11)
1179
$&$3$&$ $&$ g_?$&$ $&$ $&$ $&$ $&$ g$&$ $\\\hline
1180
$({x + {8}},11)
1181
$&$5$&$ g$&$ $&$ $&$ $&$ $&$ $&$ $&$ $\\\hline
1182
$({x^{2} + x + 1},2)
1183
$&$3$&$ $&$ g_?$&$ $&$ $&$ $&$ f_?$&$ f_?g$&$ $\\\hline
1184
$({x^{2} + x + 1},2)
1185
$&$5$&$ g$&$ $&$ $&$ $&$ $&$ f_?$&$ f_?$&$ $\\\hline
1186
$({x + {19}},31)
1187
$&$2$&$ g$&$ $&$ $&$ fg$&$ g_?$&$ $&$ $&$ g$\\\hline
1188
$({x + {19}},31)
1189
$&$3$&$ $&$ g_?$&$ $&$ f$&$ $&$ $&$ g$&$ $\\\hline
1190
$({x + {19}},31)
1191
$&$5$&$ g$&$ $&$ $&$ f$&$ $&$ $&$ $&$ $\\\hline
1192
$({x + {360}},379)
1193
$&$2$&$ g$&$ $&$ $&$ g$&$ fg_?$&$ $&$ $&$ g$\\\hline
1194
$({x + {360}},379)
1195
$&$3$&$ $&$ g_?$&$ $&$ $&$ f$&$ $&$ g$&$ $\\\hline
1196
$({x + {360}},379)
1197
$&$5$&$ g$&$ $&$ $&$ $&$ f$&$ $&$ $&$ $\\\hline
1198
\end{tabular}\end{minipage}
1199
1200
1201
1202
\noindent\begin{minipage}{\textwidth}
1203
\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=79A}$}\\
1204
\begin{tabular}{|cc|cccccccc|}\hline
1205
$P$&$Q$&869A&869B&869C&869D&869E&869F&869G&869H\\\hline
1206
$2$&$5$&$ $&$ $&$ f$&$ $&$ g_?$&$ $&$ $&$ f_?$\\\hline
1207
$2$&$7$&$ $&$ $&$ f$&$ $&$ $&$ $&$ g$&$ f_?$\\\hline
1208
$3$&$2$&$ fg$&$ $&$ $&$ g$&$ $&$ f$&$ g_?$&$ g_?$\\\hline
1209
$3$&$5$&$ f$&$ $&$ $&$ $&$ g_?$&$ f$&$ $&$ $\\\hline
1210
$3$&$7$&$ f$&$ $&$ $&$ $&$ $&$ f$&$ g$&$ $\\\hline
1211
$5$&$2$&$ g$&$ $&$ $&$ g$&$ $&$ $&$ fg_?$&$ g_?$\\\hline
1212
$5$&$7$&$ $&$ $&$ $&$ $&$ $&$ $&$ fg$&$ $\\\hline
1213
$7$&$2$&$ g$&$ $&$ $&$ g$&$ $&$ $&$ fg_?$&$ g_?$\\\hline
1214
$7$&$5$&$ $&$ $&$ $&$ $&$ g_?$&$ $&$ f$&$ $\\\hline
1215
\end{tabular}\end{minipage}
1216
1217
\noindent\begin{minipage}{\textwidth}
1218
\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=79B}$}\\
1219
\begin{tabular}{|cc|cccccccc|}\hline
1220
$P$&$Q$&869A&869B&869C&869D&869E&869F&869G&869H\\\hline
1221
$2$&$({x + {10}},13)
1222
$&$ $&$ $&$ f$&$ $&$ $&$ $&$ $&$ f_?$\\\hline
1223
$2$&$({x + {1637}},1861)
1224
$&$ $&$ $&$ f$&$ $&$ $&$ $&$ $&$ f_?g$\\\hline
1225
$2$&$({x + {3680}},5927)
1226
$&$ $&$ $&$ f$&$ $&$ $&$ g$&$ $&$ f_?$\\\hline
1227
$2$&$({x + {55}},61)
1228
$&$ $&$ $&$ f$&$ $&$ $&$ $&$ $&$ f_?g$\\\hline
1229
$3$&$({x + {10}},13)
1230
$&$ f$&$ $&$ $&$ $&$ $&$ f$&$ $&$ $\\\hline
1231
$3$&$({x + {1637}},1861)
1232
$&$ f$&$ $&$ $&$ $&$ $&$ f$&$ $&$ g$\\\hline
1233
$3$&$({x + 1},2)
1234
$&$ fg$&$ $&$ $&$ g$&$ $&$ f$&$ g_?$&$ g_?$\\\hline
1235
$3$&$({x + {3680}},5927)
1236
$&$ f$&$ $&$ $&$ $&$ $&$ fg$&$ $&$ $\\\hline
1237
$3$&$({x + {55}},61)
1238
$&$ f$&$ $&$ $&$ $&$ $&$ f$&$ $&$ g$\\\hline
1239
$5$&$({x + {10}},13)
1240
$&$ $&$ $&$ $&$ $&$ $&$ $&$ f$&$ $\\\hline
1241
$5$&$({x + {1637}},1861)
1242
$&$ $&$ $&$ $&$ $&$ $&$ $&$ f$&$ g$\\\hline
1243
$5$&$({x + 1},2)
1244
$&$ g$&$ $&$ $&$ g$&$ $&$ $&$ fg_?$&$ g_?$\\\hline
1245
$5$&$({x + {3680}},5927)
1246
$&$ $&$ $&$ $&$ $&$ $&$ g$&$ f$&$ $\\\hline
1247
$5$&$({x + {55}},61)
1248
$&$ $&$ $&$ $&$ $&$ $&$ $&$ f$&$ g$\\\hline
1249
$7$&$({x + {10}},13)
1250
$&$ $&$ $&$ $&$ $&$ $&$ $&$ f$&$ $\\\hline
1251
$7$&$({x + {1637}},1861)
1252
$&$ $&$ $&$ $&$ $&$ $&$ $&$ f$&$ g$\\\hline
1253
$7$&$({x + 1},2)
1254
$&$ g$&$ $&$ $&$ g$&$ $&$ $&$ fg_?$&$ g_?$\\\hline
1255
$7$&$({x + {3680}},5927)
1256
$&$ $&$ $&$ $&$ $&$ $&$ g$&$ f$&$ $\\\hline
1257
$7$&$({x + {55}},61)
1258
$&$ $&$ $&$ $&$ $&$ $&$ $&$ f$&$ g$\\\hline
1259
\end{tabular}\end{minipage}
1260
1261
1262
1263
\noindent\begin{minipage}{\textwidth}
1264
\vspace{.2in}\noindent{\large $\mathbf{f=19A, \, g=47A}$}\\
1265
\begin{tabular}{|cc|cccc|}\hline
1266
$P$&$Q$&893A&893B&893C&893D\\\hline
1267
$17$&$({x + {304}},1373)
1268
$&$ g$&$ $&$ $&$ f$\\\hline
1269
$17$&$({x + {37}},139)
1270
$&$ $&$ $&$ $&$ fg$\\\hline
1271
$17$&$({x^{4} + x^{3} + x^{2} + x + 1},2)
1272
$&$ $&$ g_?$&$ g$&$ f$\\\hline
1273
$17$&$({x + {20}},23)
1274
$&$ $&$ $&$ $&$ f$\\\hline
1275
$17$&$({x + 1},3)
1276
$&$ $&$ $&$ $&$ fg$\\\hline
1277
$17$&$({x + {2}},7)
1278
$&$ g_?$&$ $&$ $&$ f$\\\hline
1279
$3$&$({x + {304}},1373)
1280
$&$ g$&$ f$&$ $&$ $\\\hline
1281
$3$&$({x + {37}},139)
1282
$&$ $&$ f$&$ $&$ g$\\\hline
1283
$3$&$({x^{4} + x^{3} + x^{2} + x + 1},2)
1284
$&$ $&$ fg_?$&$ g$&$ $\\\hline
1285
$3$&$({x + {20}},23)
1286
$&$ $&$ f$&$ $&$ $\\\hline
1287
$3$&$({x + {2}},7)
1288
$&$ g_?$&$ f$&$ $&$ $\\\hline
1289
$5$&$({x + {304}},1373)
1290
$&$ g$&$ f$&$ $&$ $\\\hline
1291
$5$&$({x + {37}},139)
1292
$&$ $&$ f$&$ $&$ g$\\\hline
1293
$5$&$({x^{4} + x^{3} + x^{2} + x + 1},2)
1294
$&$ $&$ fg_?$&$ g$&$ $\\\hline
1295
$5$&$({x + {20}},23)
1296
$&$ $&$ f$&$ $&$ $\\\hline
1297
$5$&$({x + 1},3)
1298
$&$ $&$ f$&$ $&$ g$\\\hline
1299
$5$&$({x + {2}},7)
1300
$&$ g_?$&$ f$&$ $&$ $\\\hline
1301
\end{tabular}\end{minipage}
1302
1303
1304
1305
\noindent\begin{minipage}{\textwidth}
1306
\vspace{.2in}\noindent{\large $\mathbf{f=29A, \, g=31A}$}\\
1307
\begin{tabular}{|cc|cccccc|}\hline
1308
$P$&$Q$&899A&899B&899C&899D&899E&899F\\\hline
1309
$({x + {52}},113)
1310
$&$({x^{2} + x + 1},2)
1311
$&$ $&$ $&$ g$&$ $&$ g_?$&$ f$\\\hline
1312
$({x + {52}},113)
1313
$&$({x + {12}},31)
1314
$&$ $&$ $&$ $&$ $&$ $&$ fg$\\\hline
1315
$({x + {52}},113)
1316
$&$({x + {2}},5)
1317
$&$ $&$ $&$ $&$ g$&$ $&$ f$\\\hline
1318
$({x + {52}},113)
1319
$&$({x + {43}},61)
1320
$&$ $&$ $&$ $&$ g$&$ $&$ f$\\\hline
1321
$({x + {41}},47)
1322
$&$({x^{2} + x + 1},2)
1323
$&$ $&$ $&$ g$&$ $&$ fg_?$&$ $\\\hline
1324
$({x + {41}},47)
1325
$&$({x + {12}},31)
1326
$&$ $&$ $&$ $&$ $&$ f$&$ g$\\\hline
1327
$({x + {41}},47)
1328
$&$({x + {2}},5)
1329
$&$ $&$ $&$ $&$ g$&$ f$&$ $\\\hline
1330
$({x + {41}},47)
1331
$&$({x + {43}},61)
1332
$&$ $&$ $&$ $&$ g$&$ f$&$ $\\\hline
1333
$({x^{2} + {2}x + {4}},5)
1334
$&$({x^{2} + x + 1},2)
1335
$&$ $&$ $&$ g$&$ $&$ fg_?$&$ $\\\hline
1336
$({x^{2} + {2}x + {4}},5)
1337
$&$({x + {12}},31)
1338
$&$ $&$ $&$ $&$ $&$ f$&$ g$\\\hline
1339
$({x^{2} + {2}x + {4}},5)
1340
$&$({x + {43}},61)
1341
$&$ $&$ $&$ $&$ g$&$ f$&$ $\\\hline
1342
$({x + {4}},7)
1343
$&$({x^{2} + x + 1},2)
1344
$&$ $&$ $&$ g$&$ $&$ g_?$&$ $\\\hline
1345
$({x + {4}},7)
1346
$&$({x + {12}},31)
1347
$&$ $&$ $&$ $&$ $&$ $&$ g$\\\hline
1348
$({x + {4}},7)
1349
$&$({x + {2}},5)
1350
$&$ $&$ $&$ $&$ g$&$ $&$ $\\\hline
1351
$({x + {4}},7)
1352
$&$({x + {43}},61)
1353
$&$ $&$ $&$ $&$ g$&$ $&$ $\\\hline
1354
\end{tabular}\end{minipage}
1355
1356
1357
1358
\noindent\begin{minipage}{\textwidth}
1359
\vspace{.2in}\noindent{\large $\mathbf{f=17A, \, g=53A}$}\\
1360
\begin{tabular}{|cc|cccccccccc|}\hline
1361
$P$&$Q$&901A&901B&901C&901D&901E&901F&901G&901H&901I&901J\\\hline
1362
$2$&$3$&$ f$&$ $&$ fg$&$ $&$ g$&$ $&$ f_?$&$ $&$ f_?$&$ f_?$\\\hline
1363
$2$&$5$&$ fg$&$ $&$ f$&$ $&$ $&$ $&$ f_?$&$ $&$ f_?$&$ f_?$\\\hline
1364
$2$&$7$&$ f$&$ $&$ f$&$ $&$ $&$ $&$ f_?$&$ $&$ f_?$&$ f_?g$\\\hline
1365
$3$&$5$&$ g$&$ $&$ $&$ $&$ $&$ f$&$ $&$ $&$ $&$ f_?$\\\hline
1366
$3$&$7$&$ $&$ $&$ $&$ $&$ $&$ f$&$ $&$ $&$ $&$ f_?g$\\\hline
1367
$5$&$3$&$ $&$ $&$ g$&$ $&$ fg$&$ $&$ $&$ $&$ $&$ $\\\hline
1368
$5$&$7$&$ $&$ $&$ $&$ $&$ f$&$ $&$ $&$ $&$ $&$ g$\\\hline
1369
\end{tabular}\end{minipage}
1370
1371
\noindent\begin{minipage}{\textwidth}
1372
\vspace{.2in}\noindent{\large $\mathbf{f=17A, \, g=53B}$}\\
1373
\begin{tabular}{|cc|cccccccccc|}\hline
1374
$P$&$Q$&901A&901B&901C&901D&901E&901F&901G&901H&901I&901J\\\hline
1375
$2$&$({x + {47}},113)
1376
$&$ f$&$ $&$ f$&$ $&$ $&$ $&$ f_?g$&$ $&$ f_?$&$ f_?$\\\hline
1377
$2$&$({x + {10}},13)
1378
$&$ f$&$ $&$ f$&$ $&$ $&$ $&$ f_?$&$ $&$ f_?$&$ f_?$\\\hline
1379
$2$&$({x + {2078}},4139)
1380
$&$ f$&$ $&$ f$&$ $&$ $&$ $&$ f_?$&$ $&$ f_?g$&$ f_?$\\\hline
1381
$2$&$({x + {3}},5)
1382
$&$ f$&$ $&$ f$&$ $&$ $&$ g$&$ f_?$&$ $&$ f_?$&$ f_?$\\\hline
1383
$3$&$({x + {47}},113)
1384
$&$ $&$ $&$ $&$ $&$ $&$ f$&$ g$&$ $&$ $&$ f_?$\\\hline
1385
$3$&$({x + {10}},13)
1386
$&$ $&$ $&$ $&$ $&$ $&$ f$&$ $&$ $&$ $&$ f_?$\\\hline
1387
$3$&$({x + {2078}},4139)
1388
$&$ $&$ $&$ $&$ $&$ $&$ f$&$ $&$ $&$ g$&$ f_?$\\\hline
1389
$3$&$({x + {3}},5)
1390
$&$ $&$ $&$ $&$ $&$ $&$ fg$&$ $&$ $&$ $&$ f_?$\\\hline
1391
$5$&$({x + {47}},113)
1392
$&$ $&$ $&$ $&$ $&$ f$&$ $&$ g$&$ $&$ $&$ $\\\hline
1393
$5$&$({x + {10}},13)
1394
$&$ $&$ $&$ $&$ $&$ f$&$ $&$ $&$ $&$ $&$ $\\\hline
1395
$5$&$({x + {2078}},4139)
1396
$&$ $&$ $&$ $&$ $&$ f$&$ $&$ $&$ $&$ g$&$ $\\\hline
1397
\end{tabular}\end{minipage}
1398
1399
1400
1401
\noindent\begin{minipage}{\textwidth}
1402
\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=83A}$}\\
1403
\begin{tabular}{|cc|ccccccc|}\hline
1404
$P$&$Q$&913A&913B&913C&913D&913E&913F&913G\\\hline
1405
$13$&$3$&$ $&$ $&$ $&$ g$&$ fg$&$ $&$ $\\\hline
1406
$13$&$5$&$ $&$ $&$ $&$ g$&$ f$&$ $&$ $\\\hline
1407
$2$&$3$&$ $&$ f$&$ $&$ fg$&$ g$&$ $&$ $\\\hline
1408
$2$&$5$&$ $&$ f$&$ $&$ fg$&$ $&$ $&$ $\\\hline
1409
$3$&$5$&$ $&$ $&$ $&$ g$&$ f$&$ $&$ f_?$\\\hline
1410
$5$&$3$&$ $&$ $&$ $&$ g$&$ g$&$ $&$ $\\\hline
1411
\end{tabular}\end{minipage}
1412
1413
\noindent\begin{minipage}{\textwidth}
1414
\vspace{.2in}\noindent{\large $\mathbf{f=11A, \, g=83B}$}\\
1415
\begin{tabular}{|cc|ccccccc|}\hline
1416
$P$&$Q$&913A&913B&913C&913D&913E&913F&913G\\\hline
1417
$13$&$({x + {5727}},10007)
1418
$&$ $&$ $&$ g$&$ $&$ f$&$ $&$ $\\\hline
1419
$13$&$({x + {31}},197)
1420
$&$ $&$ $&$ $&$ $&$ f$&$ $&$ g$\\\hline
1421
$13$&$({x + {5}},37)
1422
$&$ $&$ $&$ $&$ $&$ f$&$ g$&$ $\\\hline
1423
$13$&$({x + {69}},379)
1424
$&$ $&$ $&$ $&$ $&$ f$&$ $&$ g$\\\hline
1425
$13$&$({x + {38}},41)
1426
$&$ $&$ $&$ $&$ $&$ f$&$ $&$ $\\\hline
1427
$13$&$({x + 1},5)
1428
$&$ g$&$ $&$ $&$ $&$ f$&$ $&$ $\\\hline
1429
$2$&$({x + {5727}},10007)
1430
$&$ $&$ f$&$ g$&$ f$&$ $&$ $&$ $\\\hline
1431
$2$&$({x + {31}},197)
1432
$&$ $&$ f$&$ $&$ f$&$ $&$ $&$ g$\\\hline
1433
$2$&$({x + {5}},37)
1434
$&$ $&$ f$&$ $&$ f$&$ $&$ g$&$ $\\\hline
1435
$2$&$({x + {69}},379)
1436
$&$ $&$ f$&$ $&$ f$&$ $&$ $&$ g$\\\hline
1437
$2$&$({x + {38}},41)
1438
$&$ $&$ f$&$ $&$ f$&$ $&$ $&$ $\\\hline
1439
$2$&$({x + 1},5)
1440
$&$ g$&$ f$&$ $&$ f$&$ $&$ $&$ $\\\hline
1441
$3$&$({x + {5727}},10007)
1442
$&$ $&$ $&$ g$&$ $&$ f$&$ $&$ f_?$\\\hline
1443
$3$&$({x + {31}},197)
1444
$&$ $&$ $&$ $&$ $&$ f$&$ $&$ f_?g$\\\hline
1445
$3$&$({x + {5}},37)
1446
$&$ $&$ $&$ $&$ $&$ f$&$ g$&$ f_?$\\\hline
1447
$3$&$({x + {69}},379)
1448
$&$ $&$ $&$ $&$ $&$ f$&$ $&$ f_?g$\\\hline
1449
$3$&$({x + {38}},41)
1450
$&$ $&$ $&$ $&$ $&$ f$&$ $&$ f_?$\\\hline
1451
$3$&$({x + 1},5)
1452
$&$ g$&$ $&$ $&$ $&$ f$&$ $&$ f_?$\\\hline
1453
$5$&$({x + {5727}},10007)
1454
$&$ $&$ $&$ g$&$ $&$ $&$ $&$ $\\\hline
1455
$5$&$({x + {31}},197)
1456
$&$ $&$ $&$ $&$ $&$ $&$ $&$ g$\\\hline
1457
$5$&$({x + {5}},37)
1458
$&$ $&$ $&$ $&$ $&$ $&$ g$&$ $\\\hline
1459
$5$&$({x + {69}},379)
1460
$&$ $&$ $&$ $&$ $&$ $&$ $&$ g$\\\hline
1461
$5$&$({x + {38}},41)
1462
$&$ $&$ $&$ $&$ $&$ $&$ $&$ $\\\hline
1463
\end{tabular}\end{minipage}
1464
1465
1466
1467
\noindent\begin{minipage}{\textwidth}
1468
\vspace{.2in}\noindent{\large $\mathbf{f=23A, \, g=41A}$} {\small (char 2 primes for $g$ omitted}\\
1469
\begin{tabular}{|cc|ccccc|}\hline
1470
$P$&$Q$&943A&943B&943C&943D&943E\\\hline
1471
$({x + {8}},11)
1472
$&$({x + {7}},13)
1473
$&$ $&$ $&$ g$&$ $&$ $\\\hline
1474
$({x + {8}},11)
1475
$&$({x + {10}},37)
1476
$&$ $&$ $&$ g$&$ $&$ $\\\hline
1477
$({x + {8}},11)
1478
$&$({x + {2}},5)
1479
$&$ $&$ $&$ $&$ $&$ $\\\hline
1480
$({x + {8}},11)
1481
$&$({x + {31}},67)
1482
$&$ $&$ $&$ $&$ $&$ g$\\\hline
1483
$({x + {124}},151)
1484
$&$({x + {7}},13)
1485
$&$ $&$ $&$ g$&$ $&$ f$\\\hline
1486
$({x + {124}},151)
1487
$&$({x + {10}},37)
1488
$&$ $&$ $&$ g$&$ $&$ f$\\\hline
1489
$({x + {124}},151)
1490
$&$({x + {2}},5)
1491
$&$ $&$ $&$ $&$ $&$ f$\\\hline
1492
$({x + {124}},151)
1493
$&$({x + {31}},67)
1494
$&$ $&$ $&$ $&$ $&$ fg$\\\hline
1495
$({x + {13}},31)
1496
$&$({x + {7}},13)
1497
$&$ $&$ $&$ g$&$ f_?$&$ $\\\hline
1498
$({x + {13}},31)
1499
$&$({x + {10}},37)
1500
$&$ $&$ $&$ g$&$ f_?$&$ $\\\hline
1501
$({x + {13}},31)
1502
$&$({x + {2}},5)
1503
$&$ $&$ $&$ $&$ f_?$&$ $\\\hline
1504
$({x + {13}},31)
1505
$&$({x + {31}},67)
1506
$&$ $&$ $&$ $&$ f_?$&$ g$\\\hline
1507
$({x + {34}},59)
1508
$&$({x + {7}},13)
1509
$&$ $&$ $&$ g$&$ f$&$ $\\\hline
1510
$({x + {34}},59)
1511
$&$({x + {10}},37)
1512
$&$ $&$ $&$ g$&$ f$&$ $\\\hline
1513
$({x + {34}},59)
1514
$&$({x + {2}},5)
1515
$&$ $&$ $&$ $&$ f$&$ $\\\hline
1516
$({