Author: William A. Stein
Compute Environment: Ubuntu 18.04 (Deprecated)
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42. I changed the Magma reference to:
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6  \bibitem{magma}
7  W.~Bosma, J.~Cannon, and C.~Playoust, \emph{The {M}agma algebra system {I}:
8    {T}he user language}, J. Symb. Comp. \textbf{24} (1997), no.~3-4, 235--265,
9     \\\protect{\sf http://www.maths.usyd.edu.au:8000/u/magma/}.
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11because David Kohel told me that this is the canonical reference.
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133. \footnote{But I checked using magma and it [$\alp - 1$] doesn't!
14             So something is wrong here, I think.}
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16> I don't know what I did wrong here.  I just re-created the character
17> in Magma and got $2\alp + 1$ instead of $\alp - 1$.  Here's the Magma
18> code:
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20>   > k<a>:=F(25);
21>   > MinimalPolynomial(a);
22>   $.1^2 + 4*$.1 + 2
23>   > (a-1)^3;
24>   a^3
25>   > Order(a-1);
26>   24
27>   > Order(a);
28>   > Order(a);
29>   24
30>   > G:=DirichletGroup(1376,k);
31>   > e:=G.3;
32>   > e;
33>   $.3 34> > e^3; 35>$.3^3
36>   > Order(e);
37>   6
38>   > e43:=e^2;
39>   > e43;
40>   \$.3^2
41>   > Order(e43);
42>   3
43>   > CRT([1,3],[2^5,43]);
44>   1121
45>   > 1121 mod 2^5;
46>   1
47>   > 1121 mod 43;
48>   3
49>   > Evaluate(e43,1121);
50>   a^8
51>   > R<x>:=PolynomialRing(F(5));
52>   > kk:=quo<R|x^2+4*x+2>;
53>   > #kk;
54>   25
55>   > x^8;
56>   x^8
57>   > kk<aa>:=quo<R|x^2+4*x+2>;
58>   > aa;
59>   aa
60>   > aa^8;
61>   2*aa + 1
62>   > Order(2*a+1);
63>   3
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65> I recreated all of the computations and using the above choice of
66> alpha appears to give exactly the same tables as in our paper.
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694.
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