Anhang A

Fundamentalsysteme

Auf den folgenden Seiten sind für einige binäre Formen mehrere Tabellen abgedruckt. Spalten entsprechen einem Grad, Zeilen einer Ordnung. Die jeweils erste Tabelle gibt eine Übersicht über die Anzahl der linear unabhängigen Kovarianten in den jeweiligen Ordgraden, die zweite Tabelle gibt Auskunft über die Anzahl der Grundformen in jedem Ordgrad. Ab der Fünfform wurde auch eine Tabelle mit dem Ergebnis der naiven Siebung abgedruckt. Bis zur binären Achtform ist angegeben, wie die jeweiligen Grundformen durch Überschiebung erzeugt werden können. Die Indizes einer Grundform K_{i,j} entsprechen der Ordnung i und dem Grad j dieser Grundform. Um die Lesbarkeit zu verbessern, wurde die jeweilige binäre Form nur mit f bezeichnet. Zudem wurde, wenn mehrere Grundformen im selben Ordgrad existieren, in ihrer Bezeichnung nicht unterschieden.

A.1 Elementare Fundamentalsysteme

A.1.1 Zweiform

Linear unabhängige Kovarianten

0 2 4 6
1 1
2 1 1
3 1 1
4 1 1
5 1 1
6 1 1

Fundamentalsystem

0 2
1 1
2 1

Grundformen

\begin{aligned}f_{0} & := f \\ f_{1} & := K_{2,0} & = (f, f)^{2} \end{aligned}

A.1.2 Dreiform

Linear unabhängige Kovarianten

0 1 2 3 4 5 6
1  1
2 1 1
3  1 1
4 1  1 1
5  1 1
6  1  2

Fundamentalsystem

0 1 2 3 1  1 2  1 3  1 4 1

Grundformen

f_{0} := f
f_{1} := K_{2,2} = (f, f)^{2}
f_{2} := K_{3,3} = (f, f_{1})^{1}
f_{3} := K_{4,0} = (f, f_{2})^{3}

A.1.3 Vierform

Linear unabhängige Kovarianten

0 2 4 6 8 10
1  1
2 1 1 1
3 1 1 1 1
4 1 2 2 1
5 1 2 1 2 1
6 2 2 1 3 1

Fundamentalsystem

0 2 4 6 1  1 2 1 1 3 1  1

Grundformen

f_{0} := f
f_{1} := K_{2,0} = (f, f)^{4}
f_{2} := K_{2,4} = (f, f)^{2}
f_{3} := K_{3,0} = (f, f_{2})^{4}
f_{4} := K_{3,6} = (f, f_{2})^{1}

A.2 Weitere klassische Fundamentalsysteme  219

A.2 Weitere klassische Fundamentalsysteme

A.2.1 Fünfform

Linear unabhängige Kovarianten

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1  1
2  1  1  1
3  1  1  1  1  1  1
4 1  2  1  2  1  2  1
5  1  1  2  2  3  2  2  2
6  2  1  4  2  4  3  4
7  1  2  3  4  4  5  4  5
8 2 1  4  3  6  5  7  5
9  1  3  5  5  7  7  8  8
10  4  3  7  6  10  8  11
11 2  4  6  8  9  11  11  13
12 3 2  7  7  11  10  15  13
13 3  5  8  10  13  14  16  17
14  6  6  12  11  17  16  21
15 3  7  10  13  16  19  20  24
16 4 4  11  11  18  18  24  23
17 4  8  13  16  20  23  27  29
18 1 9  9  18  18  26  26  33
19 5 10  15  20  24  29  32  37
20 5 6  16  17  26  27  36  36

Naive Siebung

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1  1
2  1  1
3  1  1  1
4 1  1  1
5  1  1  1  −1
6  1  1  −1  −1  −1  −1  −1
7  1  1  −1  −3  −2  −3  −2
8 1 1  −2  −3  −4  −6  −5
9  1  −1  −5  −4  −9  −8  −11
10  −1  −4  −7  −11  −14  −16
11 1  −4  −6  −14  −16  −25  −25
12 1  −2  −7  −14  −21  −32  −35
13 1 −2  −4  −15  −20  −39  −45  −59
14  −1 −3  −14 −23  −41  −54  −77
15  −2 −10 −20  −44  −60  −90  −108
16  −8  −18 −41  −62  −105  −126
17  −4 −13 −38  −61  −107  −144  −202
18 1 −3 −8  −33 −56  −110  −153  −231
19  −5 −23 −50 −103  −156  −246  −319
20  −2 −16 −39 −95  −150  −259  −340

Fundamentalsystem

0 1 2 3 4 5 6 7 8 9 # ∑
1  1  1  1
2  1  1  2  3
3  1  1  1 3  6
4 1  1  1  3  9
5  1  1  1  3 12
6  1  1  2 14
7  1  1  2 16
8 1  1  2 18
9  1  1 19
11  1  1 20
12 1  1 21
13 1 1 22
18 1  1 23

Invarianten = 4, Kovarianten = 23 Erzeugung der Grundformen

\begin{aligned}f_{1} & := K_{2,2} & = (f, f)^{4} \\ f_{2} & := K_{2,6} & = (f, f)^{2} \\ f_{3} & := K_{3,3} & = (f, f_{1})^{2} \\ f_{4} & := K_{3,5} & = (f, f_{1})^{1} \\ f_{5} & := K_{3,9} & = (f, f_{2})^{1} \\ f_{6} & := K_{4,0} & = (f, f_{4})^{5} \\ f_{7} & := K_{4,4} & = (f, f_{3})^{2} \\ f_{8} & := K_{4,6} & = (f, f_{3})^{1} \\ f_{9} & := K_{5,1} & = (f, f_{7})^{4} \\ f_{10} & := K_{5,3} & = (f, f_{7})^{3} \\ f_{11} & := K_{5,7} & = (f, f_{7})^{1} \end{aligned}  \begin{aligned}f_{12} & := K_{6,2} & = (f, f_{10})^{3} \\ f_{13} & := K_{6,4} & = (f, f_{9})^{1} \\ f_{14} & := K_{7,1} & = (f, f_{13})^{4} \\ f_{15} & := K_{7,5} & = (f, f_{12})^{1} \\ f_{16} & := K_{8,0} & = (f, f_{15})^{5} \\ f_{17} & := K_{8,2} & = (f, f_{15})^{4} \\ f_{18} & := K_{9,3} & = (f, f_{17})^{2} \\ f_{19} & := K_{11,1} & = (f, f_{9}f_{10})^{4} \\ f_{20} & := K_{12,0} & = (f, f_{10}f_{12})^{5} \\ f_{21} & := K_{13,1} & = (f, f_{1}f_{9}^{2})^{4} \\ f_{22} & := K_{18,0} & = (f, f_{1}f_{9}^{3})^{5} \end{aligned}

A.2.2 Sechsform

Linear unabhängige Kovarianten

0 2 4 6 8 10 12 14 16
1  1
2 1  1  1  1
3  1  2  1  1  1  1
4 2  2 1  3  1  3  1  2
5  2 1 4  2  4  3  4  2
6 3  4 3  6  3  7  4  6
7  4 2 7  5  8  7  9  6
8 4 1 7 5 11  7  13  9  13
9  6 4 12 9  14 13 17 13
10 6 2 11 9 17 13 22 17 23
11  9 7 18 15 23 21 28 24
12 8 4 16 14 26 21 34 28 37
13  13 11 26 23 34 33 43 38
14 10 6 23 21 37 32 49 43 56
15 1 18 16 36 33 49 48 62 57
16 13 9 31 30 51 46 68 62 80
17 1 24 23 48 46 67 67 86 81
18 16 13 41 41 68 63 92 86 109
19 2 31 31 63 62 89 90 115 111
20 20 18 53 54 88 85 120 115 145

Naive Siebung

0 2 4 6 8 10 12 14 16
1  1
2 1  1  1
3  1  1  1  1
4 1  1 1  1
5  1 1  1  −1
6 1  2 −1 −1 −1 −2 −1
7  1 1 −1  −1 −4 −2 −4
8  1  −3 −5 −5 −7 −7
9  1 −2 −4 −5 −10 −11 −15
10 1 1 −1 −2 −7 −12 −15 −22 −23
11  −4 −10 −15 −26 −29 −42
12  1 −2 −6 −15 −26 −36 −53 −60
13  −1 −9 −21 −32 −57 −67 −95
14  −4 −12 −30 −52 −71 −109 −134
15 1  −4 −16 −39 −61 −108 −141 −191
16  −7 −23 −51 −94 −136 −202 −262
17  −8 −28 −67 −108 −189 −258 −360
18  −1 −11 −38 −85 −157 −237 −355 −473
19  −14 −45 −109 −180 −315 −444 −622
20  −2 −19 −60 −132 −249 −386 −587 −808

Fundamentalsystem

0 2 4 6 8 10 12 # ∑
1  1  1  1
2 1  1  1  3  4
3  1  1 1  1  4  8
4 1  1 1  1  4 12
5  1 1  1  3 15
6 1  2  3 18
7  1 1  2 20
8  1  1 21
9  1  1 22
10 1 1  2 24
12 1 1 25
15 1  1 26

Invarianten = 5, Kovarianten = 26 Erzeugung der Grundformen

\begin{aligned}f_{1} & := K_{2,0} & = (f, f)^{6} \\ f_{2} & := K_{2,4} & = (f, f)^{4} \\ f_{3} & := K_{2,8} & = (f, f)^{2} \\ f_{4} & := K_{3,2} & = (f, f_{2})^{4} \\ f_{5} & := K_{3,6} & = (f, f_{2})^{2} \\ f_{6} & := K_{3,8} & = (f, f_{2})^{1} \\ f_{7} & := K_{3,12} & = (f, f_{3})^{1} \\ f_{8} & := K_{4,0} & = (f, f_{5})^{6} \\ f_{9} & := K_{4,4} & = (f, f_{4})^{2} \\ f_{10} & := K_{4,6} & = (f, f_{4})^{1} \\ f_{11} & := K_{4,10} & = (f, f_{5})^{1} \\ f_{12} & := K_{5,2} & = (f, f_{9})^{4} \\ f_{13} & := K_{5,4} & = (f, f_{9})^{3} \end{aligned}  \begin{aligned}f_{14} & := K_{5,8} & = (f, f_{9})^{1} \\ f_{15} & := K_{6,0} & = (f, f_{2}f_{4})^{6} \\ f_{16} & := K_{6,6} & = (f, f_{12})^{1} \\ f_{17} & := K_{6,6} & = (f, f_{13})^{2} \\ f_{18} & := K_{7,2} & = (f, f_{16})^{5} \\ f_{19} & := K_{7,4} & = (f, f_{16})^{4} \\ f_{20} & := K_{8,2} & = (f, f_{19})^{4} \\ f_{21} & := K_{9,4} & = (f, f_{20})^{2} \\ f_{22} & := K_{10,0} & = (f, f_{4}^{3})^{6} \\ f_{23} & := K_{10,2} & = (f, f_{21})^{4} \\ f_{24} & := K_{12,2} & = (f, f_{4}^{2}f_{12})^{5} \\ f_{25} & := K_{15,0} & = (f, f_{18}f_{19})^{6} \end{aligned}

A.2.3 Achtform

Linear unabhängige Kovarianten

0 2 4 6 8 10 12 14 16 18 20 22
1  1
2 1  1  1  1  1
3 1  1  1  2  1  2  1  1  1  1
4 2  3  1  4  2  4  2  4  2  3  1
5 2 1 4  3  6  5  7  5  7  5  6  4
6 4 1 7  5  11  7  13  9  13  10  12  8
7 4 3 10 9  16 14 19  17  21  18  21  17
8 7 4 16 13 25 21 31  26  35  29  35  29
9 8 8 21 22 35 33 45  42  51  48  54  48
10 12 10 32 30 51 48 66  61  77  70  83  74
11 13 17 42 45 69 70 91  90  108 105 118 112
12 20 22 58 61 96 95 128 124 152 147 169 159
13 22 33 75 85 126 133 169 173 205 205 231 226
14 31 42 101 111 168 175 227 230 277 275 314 307
15 36 59 126 150 215 233 294 307 360 369 414 414
16 47 74 165 190 279 299 381 396 471 479 543 543

Naive Siebung

0 2 4 6 8 10 12 14 16 18 20 22
1  1
2 1  1  1  1
3 1  1  1  1  1  1  1  1
4 1  2  1  1  2  1  1  1
5 1 1 2  2  1  3  1  −1  −1
6 1 1 2  3  1  1 −2 −1  −3  −4  −3  −4
7 1 2 2  3  −1 −4 −7  −7 −11 −8 −12
8 1 2 2  2 −3 −6 −12 −16 −20 −24 −26 −29
9 1 3 1  −6 −15 −23 −35 −40 −55 −54 −67
10 1 2  −4 −16 −30 −49 −64 −85 −105 −119 −136
11  2 −1 −10 −28 −56 −83 −122 −153 −202 −223 −273
12  1 −5 −20 −53 −96 −151 −210 −279 −357 −424 −500
13  −8 −38 −85 −160 −243 −360 −467 −628 −742 −901
14  −2 −18 −63 −142 −253 −403 −581 −792 −1038 −1283 −1534
15  −4 −28 −101 −215 −399 −622 −936 −1265 −1702 −2097 −2573
16 −1 −7 −49 −155 −334 −601 −973 −1443 −2013 −2674 −3392 −4147

224  Fundamentalsysteme

Fundamentalsystem

0 2 4 6 8 10 12 14 16 18 # ∑
1  1  1  1
2 1  1  1  1  4  5
3 1  1 1 1 1  1  1  1  8 13
4 1  2 1 1 2  1  1  1 10 23
5 1 1 2 2 1 3  1  11 34
6 1 1 2 3 1 1  9 43
7 1 2 2 3  8 51
8 1 2 2 2  7 58
9 1 3 1  5 63
10 1 2  3 66
11  2  2 68
12  1  1 69

Invarianten = 9, Kovarianten = 69 Erzeugung der Grundformen

\begin{aligned}f_{1} & := K_{2,0} & = (f, f)^{8} \\ f_{2} & := K_{2,4} & = (f, f)^{6} \\ f_{3} & := K_{2,8} & = (f, f)^{4} \\ f_{4} & := K_{2,12} & = (f, f)^{2} \\ f_{5} & := K_{3,0} & = (f, f_{3})^{8} \\ f_{6} & := K_{3,4} & = (f, f_{2})^{4} \\ f_{7} & := K_{3,6} & = (f, f_{2})^{3} \\ f_{8} & := K_{3,8} & = (f, f_{2})^{2} \\ f_{9} & := K_{3,10} & = (f, f_{2})^{1} \\ f_{10} & := K_{3,12} & = (f, f_{3})^{2} \\ f_{11} & := K_{3,14} & = (f, f_{3})^{1} \\ f_{12} & := K_{3,18} & = (f, f_{4})^{1} \\ f_{13} & := K_{4,0} & = (f, f_{8})^{8} \\ f_{14} & := K_{4,4} & = (f, f_{6})^{4} \\ f_{15} & := K_{4,4} & = (f, f_{7})^{5} \\ f_{16} & := K_{4,6} & = (f, f_{6})^{3} \\ f_{17} & := K_{4,8} & = (f, f_{6})^{2} \\ f_{18} & := K_{4,10} & = (f, f_{6})^{1} \\ f_{19} & := K_{4,10} & = (f, f_{7})^{2} \\ f_{20} & := K_{4,12} & = (f, f_{7})^{1} \\ f_{21} & := K_{4,14} & = (f, f_{8})^{1} \\ f_{22} & := K_{4,18} & = (f, f_{10})^{1} \\ f_{23} & := K_{5,0} & = (f, f_{17})^{8} \\ f_{24} & := K_{5,2} & = (f, f_{16})^{6} \\ f_{25} & := K_{5,4} & = (f, f_{14})^{4} \\ f_{26} & := K_{5,4} & = (f, f_{15})^{4} \\ f_{27} & := K_{5,6} & = (f, f_{14})^{3} \\ f_{28} & := K_{5,6} & = (f, f_{15})^{3} \\ f_{29} & := K_{5,8} & = (f, f_{14})^{2} \\ f_{30} & := K_{5,10} & = (f, f_{14})^{1} \\ f_{31} & := K_{5,10} & = (f, f_{15})^{1} \\ f_{32} & := K_{5,10} & = (f, f_{16})^{2} \\ f_{33} & := K_{5,14} & = (f, f_{17})^{1} \\ f_{34} & := K_{6,0} & = (f, f_{29})^{8} \end{aligned}  \begin{aligned}f_{35} & := K_{6,2} & = (f, f_{27})^{6} \\ f_{36} & := K_{6,4} & = (f, f_{25})^{4} \\ f_{37} & := K_{6,4} & = (f, f_{26})^{4} \\ f_{38} & := K_{6,6} & = (f, f_{24})^{2} \\ f_{39} & := K_{6,6} & = (f, f_{25})^{3} \\ f_{40} & := K_{6,6} & = (f, f_{26})^{3} \\ f_{41} & := K_{6,8} & = (f, f_{24})^{1} \\ f_{42} & := K_{6,10} & = (f, f_{25})^{1} \\ f_{43} & := K_{7,0} & = (f, f_{6}^{2})^{8} \\ f_{44} & := K_{7,2} & = (f, f_{38})^{6} \\ f_{45} & := K_{7,2} & = (f, f_{39})^{6} \\ f_{46} & := K_{7,4} & = (f, f_{37})^{4} \\ f_{47} & := K_{7,4} & = (f, f_{38})^{5} \\ f_{48} & := K_{7,6} & = (f, f_{35})^{2} \\ f_{49} & := K_{7,6} & = (f, f_{36})^{3} \\ f_{50} & := K_{7,6} & = (f, f_{37})^{3} \\ f_{51} & := K_{8,0} & = (f, f_{6}f_{14})^{8} \\ f_{52} & := K_{8,2} & = (f, f_{48})^{6} \\ f_{53} & := K_{8,2} & = (f, f_{49})^{6} \\ f_{54} & := K_{8,4} & = (f, f_{46})^{4} \\ f_{55} & := K_{8,4} & = (f, f_{47})^{4} \\ f_{56} & := K_{8,6} & = (f, f_{44})^{2} \\ f_{57} & := K_{8,6} & = (f, f_{45})^{2} \\ f_{58} & := K_{9,0} & = (f, f_{14}f_{15})^{8} \\ f_{59} & := K_{9,2} & = (f, f_{56})^{6} \\ f_{60} & := K_{9,2} & = (f, f_{57})^{6} \\ f_{61} & := K_{9,2} & = (f, f_{14}^{2})^{7} \\ f_{62} & := K_{9,4} & = (f, f_{55})^{4} \\ f_{63} & := K_{10,0} & = (f, f_{16}f_{24})^{8} \\ f_{64} & := K_{10,2} & = (f, f_{14}f_{24})^{6} \\ f_{65} & := K_{10,2} & = (f, f_{15}f_{24})^{6} \\ f_{66} & := K_{11,2} & = (f, f_{24}f_{25})^{6} \\ f_{67} & := K_{11,2} & = (f, f_{25}^{2})^{7} \\ f_{68} & := K_{12,2} & = (f, f_{25}f_{35})^{6} \end{aligned}

A.3 Die Siebenform  225

A.3 Die Siebenform

Linear unabhängige Kovarianten

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1  1
2  1  1  1  1
3  1  1  1  2  1  1  1
4 1  3  1  3  2  3  2
5  1  2  3  4  4  5  4  5
6  4  2  7  5  8  7  9
7  3  4  7  9  10  11  13  12
8 4  3  10  9  16  14  19  17
9  4  10  13  17  21  24  25  29
10  12  13  23  23  34  31  40
11  8  16  24  31  38  43  49  52
12 10  12  29  33  48  49  65  64
13  14  26  39  53  62  74  83  91
14 4  30  37  62  68  91  95  116
15  20  42  62  80  101  116  132  147
16 18  33  70  81  117  129  159  168
17  31  62  92  122  149  177  200  225
18 13  63  85  137  157  203  223  265
19  46  88  133  176  216  255  295  326
20 35  71  140  175  241  273  335  363
21  61  126  185  245  304  360  411  466
22 26  119  173  261  312  400  443  525
23  85  169  253  334  415  491  567  636
24 62  136  254  330  445  513  626  688
25  114  224  336  448  551  658  758  855
26 52  207  309  461  560  707  800  940
27  146  295  440  581  726  860  994  1124
28 97  236  433  565  757  888  1069  1192
29  189  378  564  750  930  1110  1281  1452
30 92  336  513  759  931  1166  1335  1560

In der Tabelle auf der übernächsten Seite sind die kleinen Zahlen von von Gall. Die Unterschiede zwischen Siebung und exakter Berechnung sind durch kursive Zahlen und hervorgehoben.

226  Fundamentalsysteme

Vollständige naive Siebung der Siebenform

0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15
1  1
2  1  1  1
3  1  1  1  1  1  1
4 1  2  1  2  1  1
5  1  2  2  2  2
6  3  2  2  2  −2  −4
7  3  2  4  2  −3  −4  −6  −8
8 3  3  3  3  −7  −6  −17  −14
9  3  5  2  −9  −12  −24  −29  −35
10  4  3  −10  −18  −42  −45  −67
11  5  3  −11  −20  −54  −73  −98  −124
12 6  6  −12  −24  −70  −96  −156  −180
13  7  −7  −24  −81  −123  −204  −271  −378
14 4  −6  −22  −86  −147  −270  −365  −557
15  3  −19  −82  −164  −316  −485  −736  −983
16 2  −6  −81  −165  −366  −580  −969  −1301
17  2  −59  −162  −379  −672  −1165  −1691  −2414
18 9  −42  −138  −386  −713  −1376  −2057  −3134
19  −11  −111  −345  −742  −1485  −2446  −3818  −5371
20 −1  −73  −304  −699  −1573  −2704  −4570  −6593
21  −29  −223  −632  −1529  −2919  −5110  −7899  −11617
22 1  −144  −520  −1435  −2926  −5599  −8983  −13974
23  −56  −385  −1237  −2816  −5736  −9941  −16013  −23813
24 −11  −238  −1004  −2542  −5695  −10400  −17914  −27540
25  −113  −694  −2157  −5295  −10522  −19023  −31014  −47967
26 −25  −437  −1663  −4705  −10102  −19597  −33482  −54414
27  −175  −1169  −3853  −9263  −19188  −34994  −59346  −92882
28 −34  −667  −2978  −7967  −18111  −34984  −62902  −102363
29  −293  −1967  −6502  −16094  −33686  −63887  −109609  −176210
30 −74  −1144  −4821  −13705  −30869  −62751  −113178  −190598

A.3 Die Siebenform  227

Naive Siebung

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 # ∑
1  1  1 1
2  1  1  1  3 4
3  1  1  1  1  1  1 6 10
4 1  2  1  2  1  1  8 18
5  1  2  2  2  2  9 27
6  3  2  2  2  9 36
7  3  2  4  2  11 47
8 3  3  3  3  12 59
9  3  5  2  10 69
10  4  3  7 76
11  5  3  8 84
12 6  6  12 96
13  7  7 103
14 4  4 107
15 3 3 110
16 2  2 112
17 2 2 114
18 9  9 123
19  0
20  0
21  0
22 1  1 124

Invarianten = 26, Kovarianten = 124

Fundamentalsystem

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 # ∑
1  1  1 1
2  1  1  1  3 4
3  1  1  1  1  1  1 6 10
4 1  2  1  2  1  1  8 18
5  1  2  2  2  2  1  10 28
6  3  2  2  2  1  10 38
7  3  2  4  2  1  12 50
8 3  3  3  3  1  13 63
9  3  5  2  1  11 74
10  4  4  1  9 83
11  5  3  1  9 92
12 6  6  1  13 105
13  7  1 1  9 114
14 4  2  6 120
15  3  1  4 124
16 2  3  5 129
17  2  2 131
18 9  0_{1}  9 140
19  1_{2}  1 141
20 1_{2}  1 142
21  0
22 2_{3}  2 144
23  1  1 145
24  0
25  0_{1}  0
26 1_{2}  1 146
27  0
28  0
29  0
30 1  1 147

Invarianten = 30_{33}, Kovarianten = 147_{153}

Erzeugung der Grundformen

\begin{aligned}f_{1} & := K_{2,2} & = (f, f)^{6} \\ f_{2} & := K_{2,6} & = (f, f)^{4} \\ f_{3} & := K_{2,10} & = (f, f)^{2} \\ f_{4} & := K_{3,3} & = (f, f_{2})^{5} \\ f_{5} & := K_{3,5} & = (f, f_{1})^{2} \\ f_{6} & := K_{3,7} & = (f, f_{1})^{1} \\ f_{7} & := K_{3,9} & = (f, f_{2})^{2} \\ f_{8} & := K_{3,11} & = (f, f_{2})^{1} \\ f_{9} & := K_{3,15} & = (f, f_{3})^{1} \\ f_{10} & := K_{4,0} & = (f, f_{6})^{7} \\ f_{11} & := K_{4,4} & = (f, f_{4})^{3} \\ f_{12} & := K_{4,4} & = (f, f_{5})^{4} \\ f_{13} & := K_{4,6} & = (f, f_{4})^{2} \\ f_{14} & := K_{4,8} & = (f, f_{4})^{1} \\ f_{15} & := K_{4,8} & = (f, f_{5})^{2} \\ f_{16} & := K_{4,10} & = (f, f_{5})^{1} \\ f_{17} & := K_{4,14} & = (f, f_{7})^{1} \\ f_{18} & := K_{5,1} & = (f, f_{13})^{6} \\ f_{19} & := K_{5,3} & = (f, f_{11})^{4} \\ f_{20} & := K_{5,3} & = (f, f_{13})^{5} \\ f_{21} & := K_{5,5} & = (f, f_{11})^{3} \\ f_{22} & := K_{5,5} & = (f, f_{12})^{3} \\ f_{23} & := K_{5,7} & = (f, f_{11})^{2} \\ f_{24} & := K_{5,7} & = (f, f_{12})^{2} \\ f_{25} & := K_{5,9} & = (f, f_{11})^{1} \\ f_{26} & := K_{5,9} & = (f, f_{12})^{1} \\ f_{27} & := K_{5,13} & = (f, f_{15})^{1} \\ f_{28} & := K_{6,2} & = (f, f_{21})^{5} \\ f_{29} & := K_{6,2} & = (f, f_{22})^{5} \\ f_{30} & := K_{6,2} & = (f, f_{23})^{6} \\ f_{31} & := K_{6,4} & = (f, f_{19})^{3} \\ f_{32} & := K_{6,4} & = (f, f_{20})^{3} \\ f_{33} & := K_{6,6} & = (f, f_{18})^{1} \\ f_{34} & := K_{6,6} & = (f, f_{19})^{2} \\ f_{35} & := K_{6,8} & = (f, f_{19})^{1} \\ f_{36} & := K_{6,8} & = (f, f_{21})^{2} \\ f_{37} & := K_{6,12} & = (f, f_{23})^{1} \\ f_{38} & := K_{7,1} & = (f, f_{33})^{6} \\ f_{39} & := K_{7,1} & = (f, f_{34})^{6} \\ f_{40} & := K_{7,1} & = (f, f_{36})^{7} \\ f_{41} & := K_{7,3} & = (f, f_{31})^{4} \\ f_{42} & := K_{7,3} & = (f, f_{32})^{4} \\ f_{43} & := K_{7,5} & = (f, f_{28})^{2} \\ f_{44} & := K_{7,5} & = (f, f_{29})^{2} \\ f_{45} & := K_{7,5} & = (f, f_{30})^{2} \\ f_{46} & := K_{7,5} & = (f, f_{31})^{3} \\ f_{47} & := K_{7,7} & = (f, f_{28})^{1} \\ f_{48} & := K_{7,7} & = (f, f_{29})^{1} \\ f_{49} & := K_{7,11} & = (f, f_{34})^{1} \\ f_{50} & := K_{8,0} & = (f, f_{47})^{7} \end{aligned}  \begin{aligned}f_{51} & := K_{8,0} & = (f, f_{48})^{7} \\ f_{52} & := K_{8,0} & = (f, f_{1}^{2}f_{4})^{7} \\ f_{53} & := K_{8,2} & = (f, f_{43})^{5} \\ f_{54} & := K_{8,2} & = (f, f_{44})^{5} \\ f_{55} & := K_{8,2} & = (f, f_{45})^{5} \\ f_{56} & := K_{8,4} & = (f, f_{41})^{3} \\ f_{57} & := K_{8,4} & = (f, f_{42})^{3} \\ f_{58} & := K_{8,4} & = (f, f_{43})^{4} \\ f_{59} & := K_{8,6} & = (f, f_{39})^{1} \\ f_{60} & := K_{8,6} & = (f, f_{40})^{1} \\ f_{61} & := K_{8,6} & = (f, f_{41})^{2} \\ f_{62} & := K_{8,10} & = (f, f_{43})^{1} \\ f_{63} & := K_{9,1} & = (f, f_{59})^{6} \\ f_{64} & := K_{9,1} & = (f, f_{60})^{6} \\ f_{65} & := K_{9,1} & = (f, f_{61})^{6} \\ f_{66} & := K_{9,3} & = (f, f_{56})^{4} \\ f_{67} & := K_{9,3} & = (f, f_{57})^{4} \\ f_{68} & := K_{9,3} & = (f, f_{58})^{4} \\ f_{69} & := K_{9,3} & = (f, f_{61})^{5} \\ f_{70} & := K_{9,3} & = (f, f_{62})^{7} \\ f_{71} & := K_{9,5} & = (f, f_{53})^{2} \\ f_{72} & := K_{9,5} & = (f, f_{54})^{2} \\ f_{73} & := K_{9,9} & = (f, f_{56})^{1} \\ f_{74} & := K_{10,2} & = (f, f_{73})^{7} \\ f_{75} & := K_{10,2} & = (f, f_{4}^{3})^{7} \\ f_{76} & := K_{10,2} & = (f, f_{1}^{2}f_{18})^{5} \\ f_{77} & := K_{10,2} & = (f, f_{1}^{2}f_{19})^{6} \\ f_{78} & := K_{10,4} & = (f, f_{66})^{3} \\ f_{79} & := K_{10,4} & = (f, f_{67})^{3} \\ f_{80} & := K_{10,4} & = (f, f_{68})^{3} \\ f_{81} & := K_{10,4} & = (f, f_{69})^{3} \\ f_{82} & := K_{10,8} & = (f, f_{66})^{1} \\ f_{83} & := K_{11,1} & = (f, f_{82})^{7} \\ f_{84} & := K_{11,1} & = (f, f_{1}f_{4}f_{18})^{6} \\ f_{85} & := K_{11,1} & = (f, f_{1}f_{4}f_{19})^{7} \\ f_{86} & := K_{11,1} & = (f, f_{19}^{2})^{6} \\ f_{87} & := K_{11,1} & = (f, f_{18}f_{21})^{6} \\ f_{88} & := K_{11,3} & = (f, f_{79})^{4} \\ f_{89} & := K_{11,3} & = (f, f_{80})^{4} \\ f_{90} & := K_{11,3} & = (f, f_{81})^{4} \\ f_{91} & := K_{11,7} & = (f, f_{74})^{1} \\ f_{92} & := K_{12,0} & = (f, f_{91})^{7} \\ f_{93} & := K_{12,0} & = (f, f_{1}^{3}f_{18})^{7} \\ f_{94} & := K_{12,0} & = (f, f_{4}^{2}f_{18})^{7} \\ f_{95} & := K_{12,0} & = (f, f_{1}f_{11}f_{18})^{7} \\ f_{96} & := K_{12,0} & = (f, f_{1}f_{4}f_{28})^{7} \\ f_{97} & := K_{12,0} & = (f, f_{21}f_{28})^{7} \\ f_{98} & := K_{12,2} & = (f, f_{91})^{6} \\ f_{99} & := K_{12,2} & = (f, f_{4}^{2}f_{19})^{7} \\ f_{100} & := K_{12,2} & = (f, f_{4}^{2}f_{20})^{7} \end{aligned}  \begin{aligned}f_{101} & := K_{12,2} & = (f, f_{1}f_{4}f_{28})^{6} \\ f_{102} & := K_{12,2} & = (f, f_{19}f_{28})^{5} \\ f_{103} & := K_{12,2} & = (f, f_{20}f_{28})^{5} \\ f_{104} & := K_{12,6} & = (f, f_{83})^{1} \\ f_{105} & := K_{13,1} & = (f, f_{104})^{6} \\ f_{106} & := K_{13,1} & = (f, f_{1}f_{18}f_{19})^{6} \\ f_{107} & := K_{13,1} & = (f, f_{1}f_{19}^{2})^{7} \\ f_{108} & := K_{13,1} & = (f, f_{1}f_{18}f_{20})^{6} \\ f_{109} & := K_{13,1} & = (f, f_{4}^{2}f_{28})^{7} \\ f_{110} & := K_{13,1} & = (f, f_{1}f_{11}f_{28})^{7} \\ f_{111} & := K_{13,1} & = (f, f_{1}f_{12}f_{28})^{7} \\ f_{112} & := K_{13,3} & = (f, f_{28}^{2})^{4} \\ f_{113} & := K_{13,5} & = (f, f_{98})^{2} \\ f_{114} & := K_{14,0} & = (f, f_{4}f_{18}f_{19})^{7} \\ f_{115} & := K_{14,0} & = (f, f_{4}f_{18}f_{20})^{7} \\ f_{116} & := K_{14,0} & = (f, f_{1}f_{19}f_{28})^{7} \\ f_{117} & := K_{14,0} & = (f, f_{1}f_{19}f_{29})^{7} \\ f_{118} & := K_{14,4} & = (f, f_{112})^{3} \\ f_{119} & := K_{14,4} & = (f, f_{113})^{4} \\ f_{120} & := K_{15,1} & = (f, f_{4}f_{18}f_{28})^{6} \\ f_{121} & := K_{15,1} & = (f, f_{4}f_{19}f_{28})^{7} \\ f_{122} & := K_{15,1} & = (f, f_{4}f_{20}f_{28})^{7} \\ f_{123} & := K_{15,3} & = (f, f_{118})^{4} \\ f_{124} & := K_{16,0} & = (f, f_{1}f_{4}f_{18}^{2})^{7} \\ f_{125} & := K_{16,0} & = (f, f_{4}f_{28}^{2})^{7} \\ f_{126} & := K_{16,2} & = (f, f_{19}^{3})^{7} \\ f_{127} & := K_{16,2} & = (f, f_{4}f_{28}^{2})^{6} \\ f_{128} & := K_{16,2} & = (f, f_{4}f_{28}f_{29})^{6} \\ f_{129} & := K_{17,1} & = (f, f_{19}^{2}f_{28})^{7} \\ f_{130} & := K_{17,1} & = (f, f_{11}f_{28}^{2})^{7} \\ f_{131} & := K_{18,0} & = (f, f_{1}f_{18}^{2}f_{19})^{7} \\ f_{132} & := K_{18,0} & = (f, f_{1}f_{18}^{2}f_{20})^{7} \\ f_{133} & := K_{18,0} & = (f, f_{19}f_{28}^{2})^{7} \\ f_{134} & := K_{18,0} & = (f, f_{20}f_{28}^{2})^{7} \\ f_{135} & := K_{18,0} & = (f, f_{19}f_{28}f_{29})^{7} \\ f_{136} & := K_{18,0} & = (f, f_{20}f_{28}f_{29})^{7} \\ f_{137} & := K_{18,0} & = (f, f_{19}f_{29}^{2})^{7} \\ f_{138} & := K_{18,0} & = (f, f_{20}f_{29}^{2})^{7} \\ f_{139} & := K_{18,0} & = (f, f_{19}f_{28}f_{30})^{7} \\ f_{140} & := K_{19,1} & = (f, f_{28}^{3})^{6} \\ f_{141} & := K_{20,0} & = (f, f_{21}f_{40}^{2})^{7} \\ f_{142} & := K_{22,0} & = (f, f_{4}f_{18}f_{28}f_{40})^{7} \\ f_{143} & := K_{22,0} & = (f, f_{1}f_{28}^{2}f_{40})^{7} \\ f_{144} & := K_{23,1} & = (f, f_{40}^{2}f_{58})^{6} \\ f_{145} & := K_{26,0} & = (f, f_{28}^{3}f_{40})^{7} \end{aligned}

Die Invariante I_{30} konnte nur nachgewiesen, aber nicht explizit berechnet werden (s. Kapitel 6.8.9).

A.4 Hohe Formen  229

A.4 Hohe Formen

A.4.1 Neunform

Fundamentalsystem bis Ordnung 20

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 # ∑
1  1 1
2  1  1  1  1  4 5
3  1  1 1 2 1 1 1 1  1 10 15
4 2  2 2 3 2 2 2 1 1  1 18 33
5  1  3  4 4 3 4 2 3  3  25 58
6  4  4 6 6 3 4  1  28 86
7  4  7  8 7 6 1 1  34120
8 5  8 10 10 4 2  39159
9  9 14 10 7 1  41200
10 5  15 15 3 1  39239
11  17 16 7 1  41280
12 14 23 4 1  42322
13  25 10 1  36358
14 17 13 1  □ ?  31389
15  26 1  ? ? ? ?  27416
16 21 3  ? ? ? ?  24440
17  7 □  7 447
18 (25)  25472
19  0
20 (2)  2 474

Invarianten = 91, Kovarianten = 474 Siebung

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 # ∑
1  1  1 1
2  1  1  1  1  4 5
3  1  1 1 2 1 1 1 1  1 10 15
4 2  2 2 3 2 2 2 1 1  1 18 33
5 1 3  4 4 3 4 2 2  23 56
6  4 4 6 6 3 3  26 82
7 4 7  8 7 5  31113
8 5 8 10 10 2  35148
9 9 14 10 2  35183
10 5 15 14  34216
11 17 16  33250
12 14 23  37287
13 25  25312
14 17 9  26338
15 26  26364
16 21  21385
17 5  5 390
18 25  25415

Invarianten = 89, Kovarianten = 415 Anmerkung: Bis Ordgrad (14,10) erfolgt die Berechnung des Fundamentalsystems mit dem Programm kovariante auf ”normale“ Weise. Wegen des zu großen Speicheraufwandes brach das Programm ab (□). Ein Neustart mit der zusätzlichen Option -t 2 (siehe Schrumpfgesetz auf Seite 205) ermöglichte es, die Berechnung bis Ordgrad (17,1) fortzusetzen. Frühere Ordgrade, in denen theoretisch noch weitere Grundformen sein könnten, wurden mit einem Fragezeichen ”?“ gekennzeichnet. Die Werte für die Invarianten in Ordnung 18 und 20 wurde mit dem Programm invariante berechnet. Nur die Anzahl, nicht aber die Invarianten selber konnten berechnet werden.

A.4.2 Zehnform

Fundamentalsystem bis Grad 12

0 2 4 6 8 10 12 14 16 18 20 22 24 26 # ∑
1  1  1 1
2 1  1  1  1  1  5 6
3  1  2 1 1 2 1 1 1 1  1  12 18
4 1  3 1 3 3 2 3 1 2 1 1  1 22 40
5  3 3 4 5 4 5 2 4  2  32 72
6 4 2 5 8 6 8 2 4  1  40 112
7  7 10 8 12 2 4  1  44 156
8 5 8 11 15 4 7  1  51 207
9 5 13 19 8 7  1  53 260
10 8 20 13 13  1  55 315
11 8 18 21  1  48 363
12 12 30 1 2  □  45 408

Invarianten = 44, Kovarianten = 408 Siebung

0 2 4 6 8 10 12 14 16 18 20 22 24 26 # ∑
1  1  1 1
2 1  1  1  1  1  5 6
3  1  2 1 1 2 1 1 1 1  1  12 18
4 1  3 1 3 3 2 3 1 2 1 1  1 22 40
5  3 3 4 5 4 5 2 4  1  31 71
6 4 2 5 8 6 8 2 3  38 109
7 7 10 8 12 2 3 42 151
8 5 8 11 15 4 5  48 199
9 5 13 19 8 4  49 248
10 8 20 12 10  50 298
11 8 18 21  47 345
12 12 30  42 387
13 15 16  31 418
14 13 17  30 448
15 19  19 467
16 5  5 472
17 3  3 475

Invarianten = 99, Kovarianten = 475

A.4.3 Elfform

Fundamentalsystem bis Grad 10

0 1 2 3 4 5 6 7 8 9 10 11 12131415161718192021222324252627282930 # ∑
1  1  1 1
2  1  1  1  1  1  5 6
3  1  1  1  2  2 1 2 1 1 1 1  1  15 21
4 2  3  2  4  3  4 3 3 3 1 2 1 1  1 33 54
5 2 4  5  7  7  6 7 5 5 3 3  2  56 110
6  7 7 11 12 11 11 5 7 1 2  1  75 185
7 8 12 18 18 18 15 6 4 1 1  1  102 287
8 10 17 25 28 21 20 3 3 1 1  1  130 417
9 21 37 37 31 18 3 3 1 1  1  153 570
10 13 45 54 31 16 3 3 1 □  166 736

Invarianten = 25, Kovarianten = 736 Siebung

0 1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930 # ∑
1  1  1 1
2  1  1  1  1  1  5 6
3  1 1 1 2 2 1 2 1 1 1 1  1  15 21
4 2  3 2 4 3 4 3 3 3 1 2 1 1  1 33 54
5  2  4 5 7 7 6 7 5 4 3 2  52 106
6  7 7 11 12 11 11 4 5  68 174
7  8 12 18 18 17 14 3  90 264
8 10  17 25 28 19 16  115 379
9  21 37 37 24 7  126 505
10 13  45 52 19  129 634
11  56 66 27  149 783
12 49  92 22  163 946
13  107 49  1561102
14 84  82  1661268
15  149  1491417
16 132  1321549
17 78 78 1627
18 192  1921819
19  0
20 12  12 1831

Invarianten = 494, Kovarianten = 1831

A.4.4 Zwölfform

Fundamentalsystem bis Grad 10

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 # ∑
1  1  1 1
2 1  1  1  1  1  1  6 7
3 1  1 1 2 1 2 2 1 2 1 1 1 1  1  18 25
4 2  3 2 4 3 4 4 3 4 2 3 1 2 1 1  1 40 65
5 2 2 5 6 7 8 6 9 5 6 3 5 1 2  1  68 133
6 4 4 9 11 12 14 10 12 3 6 1 1  87 220
7 5 10 15 20 18 21 9 9 1 1  109 329
8 7 16 24 29 21 21 1 1 1  121 450
9 9 28 33 37 15 1 1 1  125 575
10 14 39 41 30 1 □  125 700

Invarianten = 45, Kovarianten = 700 Siebung

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 # ∑
1  1  1 1
2 1  1  1  1  1  1  6 7
3 1  1 1 2 1 2 2 1 2 1 1 1 1  1  18 25
4 2  3 2 4 3 4 4 3 4 2 3 1 2 1 1  1 40 65
5 2 2 5 6 7 8 6 9 5 6 3 4 1 1  65 130
6 4 4 9 11 12 14 10 12 3 5  84 214
7 5 10 15 20 18 21 9 8  106 320
8 7 16 24 29 21 21  118 438
9 9 28 33 37 15  122 560
10 14 39 41 30  124 684
11 15 53 40  108 792
12 19 56 7  82 874
13 18 44  62 936
14 12  12 948

Invarianten = 109, Kovarianten = 948

234  Fundamentalsysteme

A.4.5 Dreizehnform

Fundamentalsystem bis Ordnung 9

0 1 2 3 4 5 6 7 8 9 1011121314151617181920212223242526272829303132333435363738 # ∑
1  1  1 1
2  1  1  1  1  1  1  6 7
3  1 1 1 2 2 2 2 2 1 2 1 1 1 1  1  21 28
4 2  4 2 5 4 5 5 5 4 4 4 2 3 1 2 1 1  1 55 83
5 3 5 7 9 11 11 10 12 8 10 6 6 4 4 1 2  1  110 193
6  11 11 18 20 20 24 17 18 9 11 2 5  1  167 360
7 14 21 32 36 38 37 34 16 13 1 1 1  244 604
8 19 33 51 62 56 61 34 17 2 1 1 337 941
9 45 82 94 93 75 □  389 1330

Invarianten = 21, Kovarianten = 1330

A.4.6 Siebzehnform

Fundamentalsystem bis Ordnung 7

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 1819202122232425262728293031323334353637383940414243444546474849505152535455 # ∑
1  1  1 1
2  1  1  1  1  1  1  1  1  8 9
3  1 1 1  2  2  2  3  3  2 3 2 2 2 2 1 2 1 1 1 1  1  36 45
4 3  5 3 7  5  8  7  9  8 8 8 8 7 6 7 4 5 3 4 2 3 1 2 1 1  1 126 171
5 4 8 12 15 18 20 23 24 22 25 21 22 18 19 14 15 10 10 7 7 3 4 2 2 1 1  1 328 499
6 1 21 23 38 42 49 56 56 61 52 54 39 43 24 30 12 16 5 7 1 3  633 1132
7 31 55 81 97 112 121 125 121 116 84 70 38 22 1 3  10772209

Invarianten = 4, Kovarianten = 2209