Sharedwww / tables / discriminants / disc.texOpen in CoCalc
Author: William A. Stein
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\documentclass[11pt]{article}
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\usepackage{amsfonts}
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\usepackage{amsmath}
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\title{Discriminants of Hecke algebras}
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\author{William A. Stein}
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\newcommand{\T}{\mathbb T}
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\newcommand{\F}{\mathbb F}
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\DeclareMathOperator{\disc}{disc}
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\begin{document}
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\maketitle
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Using an implementation of the modular symbols algorithm
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described in Cremona's book {\em Algorithms for modular
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elliptic curves} I computed, for each
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prime $N$ between $2$ and $577$, an integer $D_N$
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which is divisible by the discriminant of
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the Hecke algebra $\T_N$ associated to weight 2 cusp forms
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of level $N$ for $\Gamma_0(N)$.
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For these $N$ we give a table of factored $D_N$.
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The Hecke algebra $\T=\T_N$ is an order in a
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product $E=\prod E_t$ of totally
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real number fields. The {\em discriminant}
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$\disc(\T)$ is the product of
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the discriminants of the number fields $E_i$, multiplied by the
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square of the index of~$\T$ in its normalization.
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Fix a prime number $N$ and let $S(N)$ be the space of
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weight 2 cusp forms of level $N$ for $\Gamma_0(N)$.
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For $p$ not equal to $N$ let $T_p$ be the $p$-th Hecke
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operator, and let $d_p$ be the discriminant
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of the characteristic polynomial of $T_p$ acting on $S(N)$.
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Consider the sequence of integers
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$$d_2,\quad \gcd(d_2,d_3), \quad \gcd(d_2,d_3,d_5), \quad
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\ldots, \quad \gcd(d_2,d_3,\ldots,d_q), \quad \ldots$$
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where we omit $p$ if $p=N$.
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Since each term divides its predecessor, this sequence must
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eventually stabilize at some limit $\Delta_N$.
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Since each term is divisible by the discriminant
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of $\T$, this discriminant divides $\Delta_N$.
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I wrote a program which
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computes the above sequence until it repeats some value $D_N$
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for 15 terms. The result of that computation is given
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in table 1, which can be found at the end of this document.
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It is interesting to note that $N=389$ is the only case in our tables
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for which $N|D_N$.
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I have checked up to $N=14537$ and found no other
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cases in which this occurs. Whether or not this ever occurs
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is of interest to Ribet as this hypothesis plays a role
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in his paper, {\em Torsion points on $J_0(N)$ and galois
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representations}.
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Another problem is to determine, for each $N$ in table 1, whether
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the primes dividing $D_N$ are exactly the same as the primes dividing
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$\disc(\T)$. I have checked that this is the case for $N\leq 73$.
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If the ring $\T\otimes\F_p$ is not reduced then $p|\disc(\T)$.
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This ring can't be reduced if $T_q$ is not diagonalizable
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(modulo $p$) for some prime $q$ not equal to $N$.
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However, this sufficient condition is not always necessary as the
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case $N=37$ illustrates. Here $2$ ramifies in the Hecke algebra
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even though the Hecke operators $T_q$ with $q\not=2$
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act semisimply modulo $2$.
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%Will this difficulty will always occur
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%when there is a basis of $\Q$-rational eigenforms.
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\vfill
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\newpage
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\begin{center}
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Factored $D_N$ for $N\leq 577$.
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\begin{tabular}{|r||l|}\hline
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$N$ & $D_N$\\\hline\hline
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$11$ & $1$ \\\hline
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$13$ & $0$ \\\hline
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$17$ & $1$ \\\hline
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$19$ & $1$ \\\hline
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$23$ & $5$ \\\hline
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$29$ & $2^3$ \\\hline
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$31$ & $5$ \\\hline
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$37$ & $2^2$ \\\hline
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$41$ & $2^2.37$ \\\hline
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$43$ & $2^5$ \\\hline
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$47$ & $19.103$ \\\hline
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$53$ & $2^4.37$ \\\hline
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$59$ & $2^7.31.557$ \\\hline
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$61$ & $2^4.37$ \\\hline
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$67$ & $2^4.5^4$ \\\hline
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$71$ & $3^4.257^2$ \\\hline
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$73$ & $2^4.3^2.5.13$ \\\hline
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$79$ & $2^4.83.983$ \\\hline
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$83$ & $2^8.197.11497$ \\\hline
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$89$ & $2^6.5^3.6689$ \\\hline
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$97$ & $2^6.7^2.2777$ \\\hline
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$101$ & $2^8.17568767$ \\\hline
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\end{tabular}
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\begin{tabular}{|r||l|}\hline
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$N$ & $D_N$\\\hline\hline
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$103$ & $2^8.5.17.411721$ \\\hline
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$107$ & $2^{12}.5.7.1667.19079$ \\\hline
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$109$ & $2^{10}.7^2.7537$ \\\hline
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$113$ & $2^{10}.3^4.7^2.11^2.107$ \\\hline
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$127$ & $2^{12}.3^4.7.86235899$ \\\hline
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$131$ & $2^{19}.5.46141.75619573$ \\\hline
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$137$ & $2^{10}.5^2.29.401.895241$ \\\hline
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$139$ & $2^{14}.3^2.7^2.997.2151701$ \\\hline
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$149$ & $2^{12}.7^2.234893.1252037$ \\\hline
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$151$ & $2^{18}.7^2.11.67^2.257.439867$ \\\hline
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$157$ & $2^{13}.61.397.48795779$ \\\hline
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$163$ & $2^{15}.3^2.65657.82536739$ \\\hline
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$167$ & $2^{16}.5.8269.5103536431379173$ \\\hline
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$173$ & $2^{14}.5^2.7.29.5608385124289$ \\\hline
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$179$ & $2^{22}.3^4.7^2.313.137707.536747147$ \\\hline
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$181$ & $2^{16}.5^2.7.61.397.595051637$ \\\hline
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$191$ & $2^8.3^3.5.382146223.319500117632677$ \\\hline
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$193$ & $2^{14}.5.11^2.17.103.401.4153.680059$ \\\hline
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$197$ & $2^{18}.5^2.61.397.35217676193989$ \\\hline
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$199$ & $2^{16}.3.5^3.29.31.71^2.347.947.37316093$ \\\hline
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$211$ & $2^{20}.3.5.7^4.41^2.43.229.52184516509$ \\\hline
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$223$ & $2^{36}.7^2.19.103.3995922697473293141$ \\\hline
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\end{tabular}
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\begin{tabular}{|r||l|}\hline % tex *hack*.
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$N$ & \mbox{$D_N$\hspace{3.81in}}\\\hline\hline
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$227$ & $2^{37}.3^2.5^3.7^4.13^2.29.31^2.13591.57139.273349$ \\\hline
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$229$ & $2^{32}.107.17467.39555937.53625889$ \\\hline
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$233$ & $2^{22}.3^7.53.139.653.4127.24989.8388019$ \\\hline
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$239$ & $2^{12}.7^2.2833.51817.97423.1174779433.8920940047$ \\\hline
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$241$ & $2^{23}.97.1489.20857.651474368435017$ \\\hline
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$251$ & $2^{28}.5^2.29.373.8768135668531.2006012696666681$ \\\hline
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$257$ & $2^{65}.29.479.71711.409177.654233.32354821$ \\\hline
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$263$ & $2^{20}.11.61.397.15631853.34867513.97092067.252746489$ \\\hline
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$269$ & $2^{22}.3^2.43.151.27767.65657.5550873754172978311$ \\\hline
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$271$ & $2^{24}.3^2.1367.6091.592661.1132673.14171513.172450541$ \\\hline
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$277$ & $2^{22}.5^2.19.29.37.137^2.92767.1530091.25531570859$ \\\hline
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\hline\end{tabular}
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\begin{tabular}{|r||l|}\hline
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$N$ & $D_N$\\\hline\hline
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$281$ & $2^{22}.3.5.181.857.8388019.2647382149.1778899342669$ \\\hline
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$283$ & $2^{46}.349.1297.413713.73199099.5832488839$ \\\hline
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$293$ & $2^{26}.3^2.29.233.2351^2.69763.42711913589792108923$ \\\hline
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$307$ & $2^{50}.3^6.5^5.11^2.13^3.107^2.457.3697.21577.974513.568380457$ \\\hline
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$311$ & $2^{16}.5^2.29.3013091897.2106873009119126062143259000543887593$ \\\hline
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$313$ & $2^{24}.5.41^2.8619587.9614923.130838023.2164322751511$ \\\hline
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$317$ & $2^{26}.7.367.3217.660603043.14989400036918065702697531$ \\\hline
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$331$ & $2^{38}.3^2.53^2.229.1399.21911.205493.6363601.584461573862449$ \\\hline
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$337$ & $2^{28}.113.593.2791.2963615537.747945736667.4122851467451$ \\\hline
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$347$ & $2^{61}.5.7^2.19^2.331.349.479.617.1797330450291217.918291275915301361$ \\\hline
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$349$ & $2^{28}.13.103.1118857.72318613.6771977049413.1313981654817031$ \\\hline
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$353$ & $2^{34}.3^2.5.127^2.229.114641.551801.12611821.7779730837.24314514437$ \\\hline
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$359$ & $2^{36}.3^6.2777.16512254293.64542630435970307.2171776478013633068927$ \\\hline
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$367$ & $2^{44}.7.81421.251387.418175501.15354151381.13144405392643360366681$ \\\hline
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$373$ & $2^{32}.7.11^3.23.199.673.2143.1542194372227.72819251148518000363297$ \\\hline
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$379$ & $2^{34}.59.317.421.278329.5698591.2117788336277.2851210737989187265253$ \\\hline
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$383$ & $2^{32}.5.11^2.13.72893.3151861.16141144314299.$\\
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& $178236551484825400362837637090811$ \\\hline
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$389$ & $2^{63}.3^4.5^6.31^2.37.389.3881.215517113148241.477439237737571441$ \\\hline
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$397$ & $2^{57}.23^2.31^2.97.317.761^2.302609750073209.83566618884497478937$ \\\hline
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$401$ & $2^{96}.5^2.19.163.293^2.811.1218675071.71742740351.388881803749.$\\
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& $34393898968391$ \\\hline
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$409$ & $2^{32}.3^3.17.1667.1741.2341.537071.14884451.18631199.$\\
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& $1334964067081334453235547$ \\\hline
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$419$ & $2^{55}.17.43.113.151.167.971.493657.20375986548898473293.$\\
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& $53097073649092855361102575237$ \\\hline
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$421$ & $2^{34}.3.31.557.4729.825403.857459.144211946777593109.$\\
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& $2328579379136648917067$ \\\hline
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$431$ & $2^{91}.3^4.5^6.11.19^2.29.31.43.197.257.6947^2.37619.$\\
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& $29252013842927.806505757406715084824003$ \\\hline
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$433$ & $2^{68}.3^7.7^2.37^2.101.379.1439.3613.18719.2792477.77087971.$\\
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& $5830108671536745647$ \\\hline
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$439$ & $2^{66}.3^2.5.31^2.173.84179.85667.16794662617.$\\
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& $513841517138871835091506167235408934202857$ \\\hline
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$443$ & $2^{88}.3^2.7^2.31^2.499.6899.48508479390300197.$\\
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& $2817219327571188909266947704801865987$ \\\hline
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$449$ & $2^{40}.3.7^2.101.44933757980789.188247485945671.$\\
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& $653016225615601.1431966252229376199841$ \\\hline
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$457$ & $2^{36}.5.31^2.653.3169.38983093.52621913.33122975406370693.$\\
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& $5653726203394180386934181$ \\\hline
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\hline\end{tabular}
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\begin{tabular}{|r||l|}\hline
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$N$ & $D_N$\\\hline\hline
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$461$ & $2^{73}.5.7^2.19^3.97^2.80750473.3104029729.607263139073.$\\
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& $3729490905341009668647473177$ \\\hline
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$463$ & $2^{62}.113.311.9929.568201.132502583.1474412920219.$\\
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& $2770309905285622039024420194209857723$ \\\hline
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$467$ & $2^{71}.17^2.1212648089519.32432206859088781.$\\
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& $6296651104824906148358708614333895055221783$\\\hline
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$479$ & $2^{32}.13.17.1861.4021.28745083.41556253.1202203127423.$\\
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& $201529385024397103.7037463122648759781611869895003$\\\hline
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$487$ & $2^{72}.3^{16}.5^4.13^2.17^2.19^4.59.103^2.109.257.$\\
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& $623519211698413571686763.15408475904697077364866629$\\\hline
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$491$ & $2^{104}.5^6.19^2.43.131.479.887.5650859.$\\
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& $54796097920639362740205317747356273097682333252495603721$ \\\hline
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$499$ & $2^{69}.3^{11}.5.71^2.167.495613.25224990196319.$\\
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& $573452584782809.277143583167463430555979797274731$ \\\hline
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$503$ & $2^{78}.3^2.5^4.11^2.19^3.257.821.2003^2.13597.45587.$\\
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& $384479819.8659024393.20115672029938390602701696607766073563$ \\\hline
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$509$ & $2^{71}.3^3.13.157.971.1277.4567.3691783.42330311.$\\
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& $1157039662523351992921397.6331071860925306189417509$\\\hline
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$521$ & $2^{42}.23.53.67.929.13877.531096383.19526270957.1089951135204631559833.$\\
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& $14340527343875384245648725589439$ \\\hline
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$523$ & $2^{91}.3.5.41^3.59.149^2.1201.279121937.8371971617.$\\
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& $9059602909494267071628228952878552757512056969593$ \\\hline
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$541$ & $2^{46}.3^2.5.13.277.307.591581.1940573213.221136462575339.$\\
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& $1453183329662653.18044474614550745414465332996771$ \\\hline
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$547$ & $2^{105}.7^3.73.103^2.5501.11783.16097.43781.1152631.146768003.$\\
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& $9959758037.91268351929.102277460687.106666343972273$ \\\hline
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$557$ & $2^{46}.7^4.13^2.4787.252163.16849164271275021852893.$\\
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& $53296770296923102812608983.2381022539751738307256162767$ \\\hline
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$563$ & $2^{139}.5^2.13^4.37^2.61^2.37591.52667.155083.301703.938251.$\\
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& $46706589087295134421.299128314984453465128592656821021$ \\\hline
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$569$ & $2^{46}.73.449531828286229614392569.189316003.$\\
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& $257022598600391962761793946239.2294643649486046267496627432517$ \\\hline
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$571$ & $2^{166}.3^{12}.5^8.7^4.13^2.17.37^3.41^2.79^2.127^2.181.211.293.709.$\\
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& $1579^2.1667^2.12030433.807024744595934649052018211$ \\\hline
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$577$ & $2^{131}.3^{12}.5^4.13^3.59^2.61^2.257.163753.$\\
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& $41340850017998228328234516909328723846661.$\\
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& $85934741209775683850815667$ \\\hline
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\end{tabular}
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\end{center}
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\end{document}
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