CoCalc Public Fileswww / papers / thesis / src / notes2
Author: William A. Stein
Compute Environment: Ubuntu 18.04 (Deprecated)
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2. I changed the Magma reference to:
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\bibitem{magma}
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W.~Bosma, J.~Cannon, and C.~Playoust, \emph{The {M}agma algebra system {I}:
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{T}he user language}, J. Symb. Comp. \textbf{24} (1997), no.~3-4, 235--265,
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\\\protect{\sf http://www.maths.usyd.edu.au:8000/u/magma/}.
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because David Kohel told me that this is the canonical reference.
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3. \footnote{But I checked using magma and it [$\alp - 1$] doesn't!
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So something is wrong here, I think.}
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> I don't know what I did wrong here. I just re-created the character
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> in Magma and got $2\alp + 1$ instead of $\alp - 1$. Here's the Magma
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> code:
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> > k<a>:=F(25);
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> > MinimalPolynomial(a);
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> $.1^2 + 4*$.1 + 2
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> > (a-1)^3;
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> a^3
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> > Order(a-1);
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> 24
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> > Order(a);
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> > Order(a);
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> 24
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> > G:=DirichletGroup(1376,k);
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> > e:=G.3;
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> > e;
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> $.3 34 > > e^3; 35 >$.3^3
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> > Order(e);
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> 6
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> > e43:=e^2;
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> > e43;
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> \$.3^2
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> > Order(e43);
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> 3
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> > CRT([1,3],[2^5,43]);
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> 1121
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> > 1121 mod 2^5;
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> 1
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> > 1121 mod 43;
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> 3
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> > Evaluate(e43,1121);
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> a^8
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> > R<x>:=PolynomialRing(F(5));
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> > kk:=quo<R|x^2+4*x+2>;
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> > #kk;
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> 25
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> > x^8;
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> x^8
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> > kk<aa>:=quo<R|x^2+4*x+2>;
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> > aa;
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> aa
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> > aa^8;
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> 2*aa + 1
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> > Order(2*a+1);
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> 3
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> I recreated all of the computations and using the above choice of
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> alpha appears to give exactly the same tables as in our paper.
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4.
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