将多项式系数分解为素因子形式

青青子衿

$h=\operatorname{Hauptmod}$
\begin{array}{cc|c|ccc|c}
&N& h_N & && N & h_N\\
\hline
&2& \left[\frac{\eta(\tau)}{\eta(2\tau)}\right]^{24} &  && 9 & \left[\frac{\eta(\tau)}{\eta(9\tau)}\right]^{3}\\
\hline
&3& \left[\frac{\eta(\tau)}{\eta(3\tau)}\right]^{12} &  && 10 & \left[\frac{\eta(\tau)}{\eta(10\tau)}\right]^{3}\frac{\eta(5\tau)}{\eta(2\tau)}\\
\hline
&4& \left[\frac{\eta(\tau)}{\eta(4\tau)}\right]^{8} &  && 12 & \left[\frac{\eta(\tau)}{\eta(12\tau)}\right]^{3}\left[\frac{\eta(6\tau)}{\eta(2\tau)}\right]^{2}\frac{\eta(4\tau)}{\eta(3\tau)}\\
\hline
&5& \left[\frac{\eta(\tau)}{\eta(5\tau)}\right]^{6} &  && 13 & \left[\frac{\eta(\tau)}{\eta(13\tau)}\right]^{2}\\
\hline
&6& \left[\frac{\eta(\tau)}{\eta(6\tau)}\right]^{5}\frac{\eta(3\tau)}{\eta(2\tau)} &  && 16 & \left[\frac{\eta(\tau)}{\eta(16\tau)}\right]^{2}\frac{\eta(8\tau)}{\eta(2\tau)}\\
\hline
&7& \left[\frac{\eta(\tau)}{\eta(7\tau)}\right]^{4} &  && 18 & \left[\frac{\eta(\tau)}{\eta(18\tau)}\right]^{2}\frac{\eta(6\tau)\eta(9\tau)}{\eta(2\tau)\eta(3\tau)}\\
\hline
&8& \quad\left[\frac{\eta(\tau)}{\eta(8\tau)}\right]^{4}\left[\frac{\eta(4\tau)}{\eta(2\tau)}\right]^{2} &  && 25 & \frac{\eta(\tau)}{\eta(25\tau)}\\

\end{array}

\begin{array}{cc|c|c}
&N& j(\tau)=j_{X}(h_N)=X_{\mathscr{j}}(h) & j(N\tau)=j_{Y}(h_N)=Y_{\mathscr{j}N}(h) \\
\hline
&2& \frac{(h+256)^3}{h^2} & \frac{(h+16)^3}{h} \\
\hline
&3& \frac{(h+27) (h+243)^3}{h^3} & \frac{(h+27) (h+3)^3}{h} \\
\hline
&4& \frac{\left(h^2+256 h+4096\right)^3}{h^4 (h+16)} & \frac{\left(h^2+16 h+16\right)^3}{h (h+16)} \\
\hline
&5& \frac{\left(h^2+250 h+3125\right)^3}{h^5} & \frac{\left(h^2+10 h+5\right)^3}{h} \\
\hline
&6& \frac{(h+12)^3 \left(h^3+252 h^2+3888 h+15552\right)^3}{h^6 (h+8)^2 (h+9)^3} & \frac{(h+6)^3 \left(h^3+18 h^2+84 h+24\right)^3}{h (h+8)^3 (h+9)^2} \\
\hline
&7& \frac{\left(h^2+13 h+49\right) \left(h^2+245 h+2401\right)^3}{h^7} & \frac{\left(h^2+13 h+49\right) \left(h^2+5 h+1\right)^3}{h} \\
\hline
&8& \frac{\left(h^4+256 h^3+5120 h^2+32768 h+65536\right)^3}{h^8 (h+4) (h+8)^2} & \frac{\left(h^4+16 h^3+80 h^2+128 h+16\right)^3}{h (h+4)^2 (h+8)} \\
\hline
&9& \frac{(h+9)^3 \left(h^3+243 h^2+2187 h+6561\right)^3}{h^9 \left(h^2+9 h+27\right)} & \frac{(h+3)^3 \left(h^3+9 h^2+27 h+3\right)^3}{h \left(h^2+9 h+27\right)} \\
\hline
&10& \frac{\left(\begin{split}
&h^6+260 h^5+6400 h^4\\
&\quad+64000 h^3+320000 h^2\\
&\qquad+800000 h+800000
\end{split}\right)^3}{h^{10} (h+4)^2 (h+5)^5} & \frac{\left(\begin{split}
&h^6+20 h^5+160 h^4+640 h^3\\
&\quad+1280 h^2+1040 h+80
\end{split}\right)^3}{h (h+4)^5 (h+5)^2} \\
\hline
&12& \frac{\left(h^2+12 h+24\right)^3 \left(\begin{split}
&h^6+252 h^5+4392 h^4\\
&\quad+31104 h^3+108864 h^2\\
&\qquad+186624 h+124416
\end{split}\right)^3}{h^{12} (h+2) (h+3)^3 (h+4)^4 (h+6)^3} & \frac{\left(h^2+6 h+6\right)^3 \left(\begin{split}
&h^6+18 h^5+126 h^4+432 h^3\\
&\quad+732 h^2+504 h+24
\end{split}\right)^3}{h (h+2)^3 (h+3)^4 (h+4)^3 (h+6)} \\
\hline
&13& \frac{\left(h^2+5 h+13\right) \left(
\begin{split}
&h^4+247 h^3+3380 h^2\\
&\quad+15379 h+28561
\end{split}
\right)^3}{h^{13}} & \frac{\left(h^2+5 h+13\right) \left(
\begin{split}
&h^4+7 h^3+20 h^2+19 h+1
\end{split}
\right)^3}{h} \\
\hline
&16& \frac{\left(\begin{split}
&h^8+256 h^7+5632 h^6\\
&\quad+53248 h^5+282624 h^4\\
&\qquad+917504 h^3
+1835008 h^2\\
&\qquad\quad+2097152 h
+1048576
\end{split}
\right)^3}{h^{16} (h+2) (h+4)^4 \left(h^2+4 h+8\right)} & \frac{\left(\begin{split}
&h^8+16 h^7+112 h^6
+448 h^5\\
&\quad+1104 h^4
+1664 h^3+1408 h^2\\
&\qquad+512 h+16
\end{split}\right)^3}{h (h+2)^4 (h+4) \left(h^2+4 h+8\right)} \\
\hline
&18& \frac{\left(h^3+12 h^2+36 h+36\right)^3 \left(\begin{split}
&h^9+252 h^8\\
&\quad+4644 h^7\\
&\quad\>\>+39636 h^6\\
&\qquad+198288 h^5\\
&\qquad\>\>+629856 h^4\\
&\qquad+1294704 h^3\\
&\quad\>\>+1679616 h^2\\
&\quad+1259712 h\\
&+419904
\end{split}\right)^3}{h^{18} (h+2)^2 (h+3)^9 \left(h^2+3 h+3\right) \left(h^2+6 h+12\right)^2} & \frac{\left(h^3+6 h^2+12 h+6\right)^3 \left(
\begin{split}
&h^9+18 h^8+144 h^7\\
&\>\>+666 h^6+1944 h^5\\
&\>\>\>\>+3672 h^4+4404 h^3\\
&\>\>+3096 h^2+1008 h\\
&+24
\end{split}
\right)^3}{h (h+2)^9 (h+3)^2 \left(h^2+3 h+3\right)^2 \left(h^2+6 h+12\right)} \\
\hline
&25& \frac{\left(
\begin{split}
&h^{10}+250 h^9+4375 h^8\\
&\>\>+35000 h^7+178125 h^6\\
&\>\>\>\>+631250 h^5+1640625 h^4\\
&\>\>+3125000 h^3+4296875 h^2\\
&+3906250 h+1953125
\end{split}
\right)^3}{h^{25} \left(h^4+5 h^3+15 h^2+25 h+25\right)} & \frac{\left(
\begin{split}
&h^{10}+10 h^9+55 h^8\\
&\quad+200 h^7+525 h^6\\
&\qquad+1010 h^5+1425 h^4\\
&\quad+1400 h^3+875 h^2\\
&+250 h+5
\end{split}
\right)^3}{h \left(h^4+5 h^3+15 h^2+25 h+25\right)} \\
\end{array}

\begin{array}{cc|r}
& N & \Phi_N=\Phi_N(X,Y)=\Phi(j(\tau),j(N\tau)) \qquad\qquad\qquad\\
\hline
& 2 & (X+Y)^{3}-2^{4}\cdot3^{4}\cdot5^{3}(X+Y)^{2}+3^{3}\cdot5\cdot11XY(X+Y)\\

&&+2^{8}\cdot3^{7}\cdot5^{6}(X+Y)-X^{2}Y^{2}+3^{6}\cdot5^{3}\cdot11\cdot41XY\\
&&
-2^{12}\cdot3^{9}\cdot5^{9}\\
\hline
& 3 & (X+Y)^{4}+2^{15}\cdot3^{2}\cdot5^{3}(X+Y)^{3}\\
&&-2^{3}\cdot5\cdot23\cdot1163XY(X+Y)^{2}+2^{30}\cdot3^{3}\cdot5^{6}(X+Y)^{2}\\
&&+2^{3}\cdot3^{2}\cdot31X^{2}Y^{2}(X+Y)\\
&&+2^{15}\cdot3^{3}\cdot5^{3}\cdot23\cdot3499XY(X+Y)\\
&&+2^{45}\cdot3^{3}\cdot5^{9}(X+Y)-X^{3}Y^{3}\\
&&+2^{4}\cdot5^{3}\cdot109^{3}X^{2}Y^{2}\\
&&-2^{34}\cdot5^{9}\cdot23XY \\
\hline
& 4 & (X+Y)^{6}-2^{4}\cdot3^{7}\cdot5^{4}\cdot389(X+Y)^{5} \\
&& +3\cdot187148201XY(X+Y)^{4}\\
&& +2^{4}\cdot3^{10}\cdot5^{6}\cdot71\cdot127\cdot181243(X+Y)^{4}\\
&&-2^{4}\cdot3^{4}\cdot5\cdot17\cdot23X^{2}Y^{2}(X+Y)^{3}\\
&& +2^{4}\cdot3^{7}\cdot5^{4}\cdot19\cdot2423\cdot1186127XY(X+Y)^{3}\\
&& -2^{9}\cdot3^{16}\cdot5^{10}\cdot389\cdot1669163199(X+Y)^{3}\\
&& +2^{5}\cdot3\cdot31X^{3}Y^{3}(X+Y)^{2}\\
&& +3\cdot7^{2}\cdot9695361979399X^{2}Y^{2}(X+Y)^{2}\\
&& +2^{4}\cdot3^{10}\cdot5^{6}\cdot151\cdot84061\cdot66814421XY(X+Y)^{2}\\
&& +2^{8}\cdot3^{19}\cdot5^{12}\cdot11^{3}\cdot71\cdot127\cdot181243(X+Y)^{2}\\
&& -X^{4}Y^{4}(X+Y) +2^{5}\cdot3^{4}\cdot5\cdot1237\cdot5051X^{3}Y^{3}(X+Y)\\
&& -2^{4}\cdot3^{7}\cdot5^{4}\cdot41972804411567X^{2}Y^{2}(X+Y)\\
&& +3^{16}\cdot5^{11}\cdot11\cdot67\cdot165950081579XY(X+Y)\\
&& -2^{12}\cdot3^{25}\cdot5^{16}\cdot11^{6}\cdot389(X+Y) +2^{4}\cdot3\cdot31X^{4}Y^{4}\\
&& -2^{9}\cdot17\cdot19\cdot728628949X^{3}Y^{3}\\
&& +2^{10}\cdot3^{10}\cdot5^{6}\cdot197\cdot35537\cdot206033X^{2}Y^{2}\\
&& -2^{4}\cdot3^{19}\cdot5^{12}\cdot11^{3}\cdot67896265427XY\\
&& +2^{12}\cdot3^{27}\cdot5^{18}\cdot11^{9}\\
\hline
& 5 & (X+Y)^{6}+2^{17}\cdot3^{4}\cdot5\cdot31\cdot1193(X+Y)^{5}\\
&&-2^{2}\cdot3^{3}\cdot97\cdot2447\cdot9623XY(X+Y)^{4}\\
&&+2^{30}\cdot3^{7}\cdot5\cdot13^{2}\cdot3167\cdot204437(X+Y)^{4}\\
&&+2^{5}\cdot5^{2}\cdot13\cdot195053X^{2}Y^{2}(X+Y)^{3}\\
&&+2^{17}\cdot3^{6}\cdot5^{2}\cdot2557\cdot3083\cdot6825943XY(X+Y)^{3}\\
&&+2^{48}\cdot3^{9}\cdot5^{2}\cdot31\cdot1193\cdot24203\cdot2260451(X+Y)^{3}\\
&&-2^{2}\cdot3^{2}\cdot5\cdot131\cdot193X^{3}Y^{3}(X+Y)^{2}\\
&&+2^{4}\cdot3^{4}\cdot13\cdot22737631218337183X^{2}Y^{2}(X+Y)^{2}\\
&&-2^{31}\cdot3^{11}\cdot5\cdot6301\cdot1068941\cdot15022387XY(X+Y)^{2}\\
&&+2^{60}\cdot3^{13}\cdot5^{2}\cdot11^{3}\cdot13^{2}\cdot3167\cdot204437(X+Y)^{2}\\
&&+2^{3}\cdot3\cdot5\cdot31X^{4}Y^{4}(X+Y)\\
&&+2^{7}\cdot3^{3}\cdot5^{2}\cdot1248598635413X^{3}Y^{3}(X+Y)\\
&&+2^{19}\cdot3^{7}\cdot5^{2}\cdot727\cdot1290709586953583X^{2}Y^{2}(X+Y)\\
&&+2^{47}\cdot3^{12}\cdot5^{2}\cdot7\cdot11\cdot1447\cdot14767\cdot11608661XY(X+Y)\\
&&+2^{77}\cdot3^{16}\cdot5^{3}\cdot11^{6}\cdot31\cdot1193(X+Y)-X^{5}Y^{5}\\
&&+2^{4}\cdot5\cdot1439\cdot14471929X^{4}Y^{4}\\
&&-2^{6}\cdot3^{6}\cdot11^{2}\cdot8206111\cdot9540378607X^{3}Y^{3}\\
&&+2^{30}\cdot3^{9}\cdot5\cdot23\cdot2261239028038711711X^{2}Y^{2}\\
&&-2^{61}\cdot3^{15}\cdot11^{3}\cdot11171\cdot564000259XY\\
&&+2^{90}\cdot3^{18}\cdot5^{3}\cdot11^{9}\\
\hline
\end{array}

1. rawData = {"[0,0] 141359947154721358697753474691071362751004672000
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17. [5,1] -246683410950
18. [5,2] 2028551200
19. [5,3] -4550940
20. [5,4] 3720
21. [5,5] -1
22. [6,0] 1"};
23. triples =
24.   Map[Function[line,
25.      matches =
26.       StringCases[line,
27.        RegularExpression@"\\[(\\d+),(\\d+)\\]\\s*(-?\\d+)" -> {"$1",
28.          "$2", "$3"}];
29.      ToExpression[matches]], rawData][[1]];
30. pairs = triples[[All, 1 ;; 2]];
31. coelist = triples[[All, -1]];
32. polylist =
33.   If[#[[1]] == #[[2]], X^#[[1]]*Y^#[[2]],
34.      X^#[[1]]*Y^#[[2]] + X^#[[2]]*Y^#[[1]]] & /@ pairs;
35. coelist . polylist
37. MakeBoxes[factoredForm[e_], StandardForm] :=
38. Block[{$Factored = True}, MakeBoxes[e]]
40. Unprotect[Integer];
42. MakeBoxes[i_Integer, form_] /; $Factored :=
43. Block[{$Factored},
44.   TemplateBox[{RowBox[tosuperscript /@ FactorInteger[i]],
45.     MakeBoxes[i]}, "FactoredInteger", DisplayFunction -> (#1 &),
46.    InterpretationFunction -> (#2 &)]]
48. tosuperscript[{-1, 1}] := "-"
49. tosuperscript[{a_, 1}] := MakeBoxes[a]
50. tosuperscript[{a_, b_}] := MakeBoxes[a^b]
52. Unprotect[Power];
54. MakeBoxes[Power[a_, i_Integer], StandardForm] /; $Factored :=
55. With[{exp = Block[{$Factored}, RawBoxes@MakeBoxes[i]]},
56.   MakeBoxes[Power[a, exp]]]
58. Protect[Integer, Power];
59. (SymmetricReduction[coelist . polylist, {X, Y}, {P, Q}][[1]] //
60.    factoredForm) /. {P -> X + Y, Q -> X*Y}

Copy the Code

参考
math.berkeley.edu/~cjdowd/thesis.pdf
math.mit.edu/~drew/ClassicalModPolys.html
mathematica.stackexchange.com/questions/193804/_/193830#193830

青青子衿

1. N[(h + 256)^3/h^2 /.
2.   h -> (DedekindEta[Sqrt[2] I]/DedekindEta[2 Sqrt[2] I])^24, 50]
3. N[1728*KleinInvariantJ[Sqrt[2] I], 50]
4. N[(h + 16)^3/h /.
5.   h -> (DedekindEta[Sqrt[2] I]/DedekindEta[2 Sqrt[2] I])^24, 50]
6. N[1728*KleinInvariantJ[2 Sqrt[2] I], 50]
8. N[((h + 27) (h + 243)^3)/h^3 /.
9.   h -> (DedekindEta[Sqrt[2] I]/DedekindEta[3 Sqrt[2] I])^12, 50]
10. N[1728*KleinInvariantJ[Sqrt[2] I], 50]
11. N[((h + 27) (h + 3)^3)/h /.
12.   h -> (DedekindEta[Sqrt[2] I]/DedekindEta[3 Sqrt[2] I])^12, 50]
13. N[1728*KleinInvariantJ[3 Sqrt[2] I], 50]
15. N[(h^2 + 256 h + 4096)^3/(h^4 (h + 16)) /.
16.   h -> (DedekindEta[Sqrt[2] I]/DedekindEta[4 Sqrt[2] I])^8, 50]
17. N[1728*KleinInvariantJ[Sqrt[2] I], 50]
18. N[(h^2 + 16 h + 16)^3/(h (h + 16)) /.
19.   h -> (DedekindEta[Sqrt[2] I]/DedekindEta[4 Sqrt[2] I])^8, 50]
20. N[1728*KleinInvariantJ[4 Sqrt[2] I], 50]
22. N[(h^2 + 250 h + 3125)^3/h^5 /.
23.   h -> (DedekindEta[Sqrt[2] I]/DedekindEta[5 Sqrt[2] I])^6, 50]
24. N[1728*KleinInvariantJ[Sqrt[2] I], 50]
25. N[(h^2 + 10 h + 5)^3/h /.
26.   h -> (DedekindEta[Sqrt[2] I]/DedekindEta[5 Sqrt[2] I])^6, 50]
27. N[1728*KleinInvariantJ[5 Sqrt[2] I], 50]
29. N[((h + 12)^3 (h^3 + 252 h^2 + 3888 h + 15552)^3)/(
30.   h^6 (h + 8)^2 (h + 9)^3) /.
31.   h -> (DedekindEta[Sqrt[2] I]/
32.      DedekindEta[6 Sqrt[2] I])^5 DedekindEta[3 Sqrt[2] I]/
33.     DedekindEta[2 Sqrt[2] I], 50]
34. N[1728*KleinInvariantJ[Sqrt[2] I], 50]
35. N[((h + 6)^3 (h^3 + 18 h^2 + 84 h + 24)^3)/(h (h + 8)^3 (h + 9)^2) /.
36.   h -> (DedekindEta[Sqrt[2] I]/
37.      DedekindEta[6 Sqrt[2] I])^5 DedekindEta[3 Sqrt[2] I]/
38.     DedekindEta[2 Sqrt[2] I], 50]
39. N[1728*KleinInvariantJ[6 Sqrt[2] I], 50]
41. N[((h^2 + 13 h + 49) (h^2 + 245 h + 2401)^3)/h^7 /.
42.   h -> (DedekindEta[Sqrt[2] I]/DedekindEta[7 Sqrt[2] I])^4, 50]
43. N[1728*KleinInvariantJ[Sqrt[2] I], 50]
44. N[((h^2 + 13 h + 49) (h^2 + 5 h + 1)^3)/h /.
45.   h -> (DedekindEta[Sqrt[2] I]/DedekindEta[7 Sqrt[2] I])^4, 50]
46. N[1728*KleinInvariantJ[7 Sqrt[2] I], 50]
48. N[(h^4 + 256 h^3 + 5120 h^2 + 32768 h + 65536)^3/(
49.   h^8 (h + 4) (h + 8)^2) /.
50.   h -> (DedekindEta[Sqrt[2] I]/
51.      DedekindEta[8 Sqrt[2] I])^4 (DedekindEta[4 Sqrt[2] I]/
52.      DedekindEta[2 Sqrt[2] I])^2, 50]
53. N[1728*KleinInvariantJ[Sqrt[2] I], 50]
54. N[(h^4 + 16 h^3 + 80 h^2 + 128 h + 16)^3/(h (h + 4)^2 (h + 8)) /.
55.   h -> (DedekindEta[Sqrt[2] I]/
56.      DedekindEta[8 Sqrt[2] I])^4 (DedekindEta[4 Sqrt[2] I]/
57.      DedekindEta[2 Sqrt[2] I])^2, 50]
58. N[1728*KleinInvariantJ[8 Sqrt[2] I], 50]
60. N[((h + 9)^3 (h^3 + 243 h^2 + 2187 h + 6561)^3)/(
61.   h^9 (h^2 + 9 h + 27)) /.
62.   h -> (DedekindEta[Sqrt[2] I]/DedekindEta[9 Sqrt[2] I])^3, 50]
63. N[1728*KleinInvariantJ[Sqrt[2] I], 50]
64. N[((h + 3)^3 (h^3 + 9 h^2 + 27 h + 3)^3)/(h (h^2 + 9 h + 27)) /.
65.   h -> (DedekindEta[Sqrt[2] I]/DedekindEta[9 Sqrt[2] I])^3, 50]
66. N[1728*KleinInvariantJ[9 Sqrt[2] I], 50]
68. N[(h^6 + 260 h^5 + 6400 h^4
69.     + 64000 h^3 + 320000 h^2
70.     + 800000 h + 800000)^3/(h^10 (h + 4)^2 (h + 5)^5) /.
71.   h -> (DedekindEta[Sqrt[2] I]/
72.      DedekindEta[10 Sqrt[2] I])^3 DedekindEta[5 Sqrt[2] I]/
73.     DedekindEta[2 Sqrt[2] I], 50]
74. N[1728*KleinInvariantJ[Sqrt[2] I], 50]
75. N[(h^6 + 20 h^5 + 160 h^4
76.     + 640 h^3 + 1280 h^2
77.     + 1040 h + 80)^3/(h (h + 4)^5 (h + 5)^2) /.
78.   h -> (DedekindEta[Sqrt[2] I]/
79.      DedekindEta[10 Sqrt[2] I])^3 DedekindEta[5 Sqrt[2] I]/
80.     DedekindEta[2 Sqrt[2] I], 50]
81. N[1728*KleinInvariantJ[10 Sqrt[2] I], 50]
83. N[((h^2 + 12 h + 24)^3 (h^6 + 252 h^5 + 4392 h^4
84.      + 31104 h^3 + 108864 h^2
85.      + 186624 h + 124416)^3)/(
86.   h^12 (h + 2) (h + 3)^3 (h + 4)^4 (h + 6)^3) /.
87.   h -> (DedekindEta[Sqrt[2] I]/
88.      DedekindEta[12 Sqrt[2] I])^3 (DedekindEta[6 Sqrt[2] I]/
89.      DedekindEta[2 Sqrt[2] I])^2 DedekindEta[4 Sqrt[2] I]/
90.     DedekindEta[3 Sqrt[2] I], 50]
91. N[1728*KleinInvariantJ[Sqrt[2] I], 50]
92. N[((h^2 + 6 h + 6)^3 (h^6 + 18 h^5 + 126 h^4
93.      + 432 h^3 + 732 h^2
94.      + 504 h + 24)^3)/(h (h + 2)^3 (h + 3)^4 (h + 4)^3 (h + 6)) /.
95.   h -> (DedekindEta[Sqrt[2] I]/
96.      DedekindEta[12 Sqrt[2] I])^3 (DedekindEta[6 Sqrt[2] I]/
97.      DedekindEta[2 Sqrt[2] I])^2 DedekindEta[4 Sqrt[2] I]/
98.     DedekindEta[3 Sqrt[2] I], 50]
99. N[1728*KleinInvariantJ[12 Sqrt[2] I], 50]
101. N[((h^2 + 5 h + 13) (h^4
102.      + 247 h^3 + 3380 h^2
103.      + 15379 h + 28561)^3)/h^13 /.
104.   h -> (DedekindEta[Sqrt[2] I]/DedekindEta[13 Sqrt[2] I])^2, 50]
105. N[1728*KleinInvariantJ[Sqrt[2] I], 50]
106. N[((h^2 + 5 h + 13) (h^4
107.      + 7 h^3 + 20 h^2
108.      + 19 h + 1)^3)/h /.
109.   h -> (DedekindEta[Sqrt[2] I]/DedekindEta[13 Sqrt[2] I])^2, 50]
110. N[1728*KleinInvariantJ[13 Sqrt[2] I], 50]
112. N[(h^8 + 256 h^7 + 5632 h^6 + 53248 h^5
113.     + 282624 h^4 + 917504 h^3
114.     + 1835008 h^2 + 2097152 h
115.     + 1048576)^3/(h^16 (h + 2) (h + 4)^4 (h^2 + 4 h + 8)) /.
116.   h -> (DedekindEta[Sqrt[2] I]/
117.      DedekindEta[16 Sqrt[2] I])^2 DedekindEta[8 Sqrt[2] I]/
118.     DedekindEta[2 Sqrt[2] I], 50]
119. N[1728*KleinInvariantJ[Sqrt[2] I], 50]
120. N[(h^8 + 16 h^7 + 112 h^6 + 448 h^5
121.     + 1104 h^4 + 1664 h^3
122.     + 1408 h^2 + 512 h
123.     + 16)^3/(h (h + 2)^4 (h + 4) (h^2 + 4 h + 8)) /.
124.   h -> (DedekindEta[Sqrt[2] I]/
125.      DedekindEta[16 Sqrt[2] I])^2 DedekindEta[8 Sqrt[2] I]/
126.     DedekindEta[2 Sqrt[2] I], 50]
127. N[1728*KleinInvariantJ[16 Sqrt[2] I], 50]
129. N[((h^3 + 12 h^2 + 36 h + 36)^3 (h^9 + 252 h^8 + 4644 h^7 + 39636 h^6
130.      + 198288 h^5 + 629856 h^4 + 1294704 h^3 + 1679616 h^2
131.      + 1259712 h + 419904)^3)/(
132.   h^18 (h + 2)^2 (h + 3)^9 (h^2 + 3 h + 3) (h^2 + 6 h + 12)^2) /.
133.   h -> (DedekindEta[Sqrt[2] I]/DedekindEta[18 Sqrt[2] I])^2 (
134.     DedekindEta[6 Sqrt[2] I] DedekindEta[9 Sqrt[2] I])/(
135.     DedekindEta[2 Sqrt[2] I] DedekindEta[3 Sqrt[2] I]), 50]
136. N[1728*KleinInvariantJ[Sqrt[2] I], 50]
137. N[((h^3 + 6 h^2 + 12 h + 6)^3 (h^9 + 18 h^8 + 144 h^7 + 666 h^6
138.      + 1944 h^5 + 3672 h^4 + 4404 h^3 + 3096 h^2
139.      + 1008 h + 24)^3)/(
140.   h (h + 2)^9 (h + 3)^2 (h^2 + 3 h + 3)^2 (h^2 + 6 h + 12)) /.
141.   h -> (DedekindEta[Sqrt[2] I]/DedekindEta[18 Sqrt[2] I])^2 (
142.     DedekindEta[6 Sqrt[2] I] DedekindEta[9 Sqrt[2] I])/(
143.     DedekindEta[2 Sqrt[2] I] DedekindEta[3 Sqrt[2] I]), 50]
144. N[1728*KleinInvariantJ[18 Sqrt[2] I], 50]
146. N[(h^10 + 250 h^9 + 4375 h^8
147.     + 35000 h^7 + 178125 h^6
148.     + 631250 h^5 + 1640625 h^4
149.     + 3125000 h^3 + 4296875 h^2
150.     + 3906250 h + 1953125)^3/(
151.   h^25 (h^4 + 5 h^3 + 15 h^2 + 25 h + 25)) /.
152.   h -> DedekindEta[Sqrt[2] I]/DedekindEta[25 Sqrt[2] I], 50]
153. N[1728*KleinInvariantJ[Sqrt[2] I], 50]
154. N[(h^10 + 10 h^9 + 55 h^8
155.     + 200 h^7 + 525 h^6
156.     + 1010 h^5 + 1425 h^4
157.     + 1400 h^3 + 875 h^2
158.     + 250 h + 5)^3/(h (h^4 + 5 h^3 + 15 h^2 + 25 h + 25)) /.
159.   h -> DedekindEta[Sqrt[2] I]/DedekindEta[25 Sqrt[2] I], 50]
160. N[1728*KleinInvariantJ[25 Sqrt[2] I], 50]

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