Discriminants of Hecke Algebras

William A. Stein^{∗}

January 28, 1999

Abstract

For N < 250 we compute the discriminants of the Hecke algebras associated to weight 2 cusp forms and newforms of level N and determine the primes p so that the cusp ∞ is a Weierstrass point on X_{0}(N)/F_{p}.

1 Introduction

Let S_{2}(N) be the space of weight 2 cusp forms for Γ_{0}(N). The Hecke algebra T ⊂ End(S_{2}(N)) is a finite commutative Z-algebra. Its discriminant Δ = disc(T) is important in studying congruences between modular forms. Let Δ^{new} be the discriminant of the new Hecke algebra T^{new} ⊂ End(S_{2}^{new}(N)). By “multiplicity one” T^{new} is a subring of a product of fields and T̃^{new} is the product of the rings of integers of these fields.

Let W be the sub Z-module of T generated by the Hecke operator T_{1}, T_{2}, … T_{g} where g is the genus of X_{0}(N). When N is prime W has finite index in T, but when ∞ is a Weierstrass point on X_{0}(N) the index will not be finite. If p ∤ N, the cusp ∞ is a Weierstrass point on X_{0}(N)/F_{p} iff p|[T : W].

2 Tables

The notation is as above.

∗University of California, Berkeley, U.S.A., was@math.berkeley.edu

N  Δ^{new}  Δ/Δ^{new}  [T̃^{new} : T^{new}] [T : W]
11  1  1  1  1
14  1  1  1  1
15  1  1  1  1
17  1  1  1  1
19  1  1  1  1
20  1  1  1  1
21  1  1  1  1
23  5  1  1  1
24  1  1  1  1
26  2^{2}  1  2  1
27  1  1  1  1
29  2^{3}  1  1  1
30  1  −2^{2}.7  1  1
31  5  1  1  1
32  1  1  1  1
33  1  −3^{2}.11  1  1
34  1  −2^{2}.7  1  1
35  2^{2}.17  1  2  1
36  1  1  1  1
37  2^{2}  1  2  1
38  2^{2}  −2^{3}.3^{2}  2  1
39  2^{5}  1  2  1
40  1  0  1  ?
41  2^{2}.37  1  1  1
42  1  2^{9}.7  1  1
43  2^{5}  1  2  2
44  1  −2^{6}  1  1
45  1  2^{2}  1  1
46  1  2^{2}.5^{4}.41  1  1
47  19.103  1  1  1
48  1  0  1  ?
49  1  1  1  1
50  2^{2}  1  2  1
51  2^{2}.17  −2^{6}.3  2  3
52  1  2^{4}.3^{2}  1  3
53  2^{4}.37  1  2  3
54  2^{2}  −2^{3}.3^{4}  2  ∞
55  2^{5}  −7^{2}.19  2  1
56  2^{2}  0  2  ?
57  2^{2}.3^{2}  −2^{7}  2.3  1
58  2^{2}  2^{10}.17  2  2
59 2^{3}.31.557  1  1  1
61  2^{4}.37  1  2  2
62  2^{4}.3  2^{2}.5^{2}.11^{2}.41  2  3
63  2^{4}.3  2^{4}  2  2
64  1  0  1  ∞
65  2^{11}.3  1  2^{3}  3
66  2^{4}  −2^{8}.3^{4}.5^{2}.7.11^{2}  2^{2}  2
67  2^{4}.5^{4}  1  2^{2}.5  2^{3}
68  2^{2}.3  −2^{8}.3^{2}.7  1  2^{2}
69  2^{4}.5  5^{2}.7^{2}.11^{2}  2^{2}  2
70  1  2^{18}.3^{2}.5.17^{2}  1  3
71 3^{4}.257^{2}  1  3^{2}  1

N  Δ^{new}  Δ/Δ^{new}  [T̃^{new} : T^{new}] [T : W]
72  1  0  1  ∞
73  2^{4}.3^{2}.5.13  1  2^{2}.3  2
74  2^{4}.5.13  2^{9}.3^{2}.5^{2}  2^{2}  11
75  2^{2}.3^{2}  2^{2}.3^{2}  2.3  1
76  1  −2^{11}.3^{6}  1  1
77  2^{6}.5^{3}  −2^{7}.3^{3}  2^{3}.5  5
78  1  −2^{22}.3.5^{2}.7^{3}.11.17  1  2^{2}
79  2^{2}.83.983  1  2  2^{3}
80  2^{2}  0  2  ?
81  2^{2}.3  0  1  ?
82  2^{5}  −2^{10}.23.37^{2}  2  2
83  2^{4}.197.11497  1  2  2^{2}
84  2^{2}  −2^{22}.3^{4}.7  2  3
85  2^{11}.3  −2^{10}  2^{3}  7
86  2^{4}.3.5.7  −2^{14}.3^{2}.5^{2}.7^{2}  2^{2}  41
87  2^{4}.5.229  2^{6}.23^{2}.73  2^{2}  2^{3}
88  2^{6}.17  0  2^{3}  ?
89  2^{6}.5^{3}.6689  1  2.5  3^{2}
90  2^{4}  −2^{22}.7^{3}  2^{2}  1
91  2^{15}.79  1  2^{5}  2
92  2^{2}  2^{12}.3^{2}.5^{7}.41  2  1
93  2^{4}.5.229  2^{12}.5^{2}  2^{2}  2^{3}
94  2^{5}  2^{4}.19^{2}.47^{2}.103^{2}.457  2  7
95  2^{12}.37.709  −3^{4}.11  2^{3}  11
96  2^{2}  0  2  ∞
97  2^{6}.7^{2}.2777  1  2^{3}  2.3
98  2^{5}  −2^{12}.7  2  ∞
99  2^{4}.3^{4}  −2^{6}.3^{8}.11  2^{2}.3^{2}  2
100  1  2^{8}.3^{4}  1  ∞
101  2^{8}.17568767  1  2  3^{3}
102  2^{4}  2^{41}.3^{6}.7^{2}.17^{2}  2^{2}  3.5^{2}
103  2^{4}.5.17.411721  1  2^{2}  2^{2}.7
104  2^{2}.17  0  2  ?
105  2^{4}.5  −2^{30}.5^{2}.7.11.13.17^{2}  2^{2}  2.3
106  2^{4}.3^{2}  2^{18}.5^{2}.7.37^{2}.151  2^{2}.3  2.13
107 2^{6}.5.7.1667.19079  1  2^{2}  2^{2}.5
108  1  0  1  ∞
109  2^{10}.7^{2}.7537  1  2^{5}  2.5
110  2^{8}.3.11  −2^{22}.5^{2}.7^{7}.17.19^{2}  2^{4}  2.3
111  2^{12}.37.389  2^{4}.3.5^{2}.7^{2}.11  2^{3}  2^{2}.3
112  2^{6}  0  2^{3}  ?
113 2^{10}.3^{4}.7^{2}.11^{2}.107  1  2^{4}.3.11  2^{2}.17
114  2^{4}  −2^{36}.3^{12}.5^{6}.7.11^{2}  2^{2}  3^{5}
115  2^{6}.5.17^{2}.53  2^{12}.5^{4}.11  2^{3}  2^{2}
116  2^{4}.5^{2}  2^{29}.3^{4}.17  2^{2}.5  2^{2}.7
117  2^{11}.3  2^{24}  2^{3}  2^{4}
118  2^{4}.3^{2}  −2^{18}.7.19^{2}.31^{2}.557^{2}  2^{2}.3  223
119 2^{8}.71.131.311.1459  −2^{6}.3^{3}  2^{4}  2^{4}
120  2^{2}  0  2  ∞
121  2^{4}.3^{2}  2^{6}.3^{2}  2^{2}.3  ∞
122  2^{6}.13.229  2^{18}.7.13^{2}.37^{2}.151  2^{3}  2.83
123  2^{15}.79  −2^{8}.5^{2}.23^{2}.37^{2}.191  2^{5}  13
124  2^{2}  2^{20}.3^{6}.5^{3}.11^{4}.41  2  3^{2}

N  Δ^{new}  Δ/Δ^{new}  [T̃^{new} : T^{new}] [T : W]
125  2^{8}.5^{12}.11  0  2^{2}.5^{4}  ∞
126  2^{2}  2^{49}.3^{8}.5^{2}.7^{3}  2  2
127  2^{6}.3^{4}.7.86235899  1  2^{3}  2^{2}.29
128  2^{16}  0  2^{8}  ∞
129  2^{12}.5^{2}.71  −2^{23}.3^{2}.5^{2}.7^{2}  2^{3}.5  17
130  2^{4}  −2^{38}.3^{6}.5^{4}.7^{3}.11.17.19  2^{2}  5.7
131  2^{9}.5.46141.75619573  1  2  2.5^{2}
132  2^{2}  −2^{32}.3^{12}.5^{6}.7.11^{4}  2  2^{3}
133  2^{16}.5^{2}.13.229  −3^{5}.7^{2}  2^{8}  2.43
134  2^{6}.3^{4}.11.43  −2^{10}.3^{2}.5^{12}.19^{2}.41  2^{3}  29.31
135  2^{6}.3^{4}.13^{2}  0  2^{3}.3^{2}  ∞
136  2^{8}.5  0  2^{4}  ?
137  2^{10}.5^{2}.29.401.895241  1  2^{4}  2^{5}.5
138  2^{8}.5  2^{32}.3^{3}.5^{10}.7^{5}.11^{6}.41^{2}  2^{4}  2^{4}.7^{2}
139  2^{8}.3^{2}.7^{2}.997.2151701  1  2^{4}.3  2^{3}.41
140  2^{2}  2^{43}.3^{11}.5^{4}.17^{3}  2  2^{2}.3
141  2^{14}.3^{2}.17  3^{2}.7^{2}.19^{2}.43^{2}.103^{2}.3529  2^{7}.3  2^{2}.5.13
142  2^{8}.3^{2}  2^{6}.3^{22}.7.103.257^{4}  2^{4}.3  3.67
143  2^{10}.5.7.19.103.5560463  −2^{6}.3^{6}  2^{5}  2.3^{2}
144  2^{2}  0  2  ∞
145  2^{21}.37^{2}  2^{6}.5^{4}.7^{2}.19^{2}  2^{7}  2.7
146  2^{12}.101.389  −2^{12}.3^{10}.5^{2}.7.13^{2}.17.19^{2}  2^{3}  97
147  2^{14}.3^{2}.7^{4}  −2^{14}.3^{3}  2^{4}.3.7^{2}  ∞
148  2^{2}.17  2^{27}.3^{10}.5^{6}.13^{2}  2  3^{2}.11^{2}
149  2^{12}.7^{2}.234893.1252037  1  2^{3}  107
150  2^{4}  −2^{34}.3^{12}.5^{8}.7^{3}.11^{2}  2^{2}  ∞
151  2^{6}.7^{2}.11.67^{2}.257.439867  1  2^{3}.67  2.11^{2}
152  2^{8}.31^{2}  0  2^{4}  ?
153  2^{12}.3^{2}.17  −2^{30}.3^{5}.17^{2}  2^{6}.3  3^{3}
154  2^{8}.5  2^{56}.3^{14}.5^{8}.7.11  2^{4}  2.23
155  2^{24}.29.73.5077  5^{4}.7^{4}.19^{2}  2^{10}  2.73
156  2^{2}  2^{57}.3^{11}.5^{4}.7^{5}.11^{2}.17  2  2^{2}.3
157  2^{13}.61.397.48795779  1  2^{5}  2.13
158  2^{13}.3^{3}.5^{2}  2^{22}.7.17.53^{2}.83^{2}.271.983^{2}  2^{5}.3.5  2^{2}.541
159  2^{8}.19.103.1054013  2^{8}.3.7^{2}.37^{2}.107^{2}.227  2^{4}  2^{4}.3.5
160  2^{13}  0  2^{5}  ∞
161  2^{16}.5^{3}.37.536777  2^{12}.5^{2}.19^{2}.29  2^{6}.5  2.97
162  2^{4}.3^{4}  0  2^{2}.3^{2}  ∞
163  2^{15}.3^{2}.65657.82536739  1  2^{6}.3  2^{2}.347
164  2^{4}.1613  −2^{40}.3^{6}.23.37^{3}  1  2^{3}.5
165  2^{17}.3.37  −2^{50}.3^{5}.5^{2}.7^{5}.11^{2}.19^{2}  2^{5}  2^{2}.3
166  2^{6}.5.229  −2^{26}.7.71.131^{2}.197^{2}.11497^{2}  2^{3}  2.5.479
167 2^{4}.5.8269.5103536431379173  1  2^{2}  2.3^{2}
168  2^{2}  0  2  ∞
169  2^{8}.3.7^{4}.13^{4}  1  2^{3}.13^{2}  ∞
170  2^{12}.5^{2}.17  2^{68}.3^{8}.5^{3}.7^{5}.17  2^{6}.5  839
171  2^{16}.3^{7}.11^{2}  −2^{33}.3^{8}  2^{7}.3^{2}  2.3^{3}
172  2^{5}  −2^{39}.3^{10}.5^{6}.7^{6}  2  41^{2}
173 2^{14}.5^{2}.7.29.5608385124289  1  2^{4}  79
174  2^{8}.5^{2}  −2^{38}.3^{3}.5^{4}.7^{4}.11^{3}.13^{2}.17^{2} …  2^{4}.5  2^{6}.5.11^{2}

… 23^{4}.41.47.73^{2}.229^{2}

175  2^{12}.3^{4}.5^{6}.17  2^{18}.3^{4}.17^{2}  2^{6}.3^{2}.5^{2}  2^{2}
176  2^{14}.17  0  2^{7}  ?

N  Δ^{new}  Δ/Δ^{new}  [T̃^{new} : T^{new}] [T : W]
177  2^{16}.5^{3}.229  −2^{6}.31^{4}.229^{2}.557^{2}.20627  2^{8}  2^{4}.149
178  2^{16}.71  −2^{30}.5^{6}.7^{4}.199.6689^{2}  2^{5}  2^{2}.487
179  2^{12}.3^{4}.7^{2}.313.137707.536747147  1  2^{3}.3^{2}  2.59
180  1  −2^{73}.3^{16}.5.7^{3}  1  ∞
181  2^{16}.5^{2}.7.61.397.595051637  1  2^{5}  2^{4}.3^{3}
182  2^{8}.3^{2}  2^{51}.3^{20}.5^{4}.7^{6}.11^{2}.23.79^{2}  2^{4}.3  2^{2}.3^{2}.5.37
183  2^{22}.37.127.5623  2^{27}.3^{2}.19^{2}.23.37^{2}  2^{5}  2^{2}.5^{4}
184  2^{14}.3^{2}.17  0  2^{7}.3  ?
185  2^{26}.3^{2}.23029.60869  2^{24}.3^{2}.5  2^{9}.3  193
186  2^{8}.17  2^{62}.3^{5}.5^{8}.7^{2}.11^{6}.13.19^{2} …  2^{4}  2^{2}.5^{5}
… 41^{2}.127.229^{2}
187  2^{28}.3.5^{2}.17.37.8461  2^{24}.3^{4}.11  2^{11}.5  2.3^{2}.11
188  2^{4}.5.13  2^{34}.3^{6}.19^{3}.47^{4}.103^{3}.457  2^{2}  7^{2}
189  2^{16}.3^{9}.7  0  2^{6}.3^{4}  ∞
190  2^{8}.17  −2^{56}.3^{16}.5^{3}.11^{4}.13^{2}.37^{2} …  2^{4}  11.53
… 113.151.709^{2}
191 2^{4}.3^{3}.5.382146223.319500117632677  1  2^{2}  2^{2}.223
192  2^{12}  0  2^{6}  ∞
193  2^{14}.5.11^{2}.17.103.401.4153.680059  1  2^{7}.11  2^{7}
194  2^{20}.7^{2}.137.223  −2^{14}.7^{5}.67^{2}.71^{2}.433.2777^{2}  2^{4}.7  3.22283
195  2^{14}.3^{2}.79  −2^{77}.3^{9}.5^{2}.7^{6}.11^{2}.13  2^{6}.3  2.3^{2}.5.7
196  2^{5}.7^{4}  −2^{38}.3^{8}.7  2.7^{2}  ∞
197  2^{18}.5^{2}.61.397.35217676193989  1  2^{6}.5  7.383
198  2^{10}  −2^{78}.3^{24}.5^{6}.7^{5}.11^{2}  2^{5}  2.7^{2}
199 2^{8}.3.5^{3}.29.31.71^{2}.347.947.37316093  1  2^{4}.71  2^{4}.241
200  2^{8}.3^{2}.5^{4}  0  2^{4}.3.5^{2}  ∞
201  2^{18}.3^{2}.37.269.953  −2^{15}.3^{2}.5^{10}.19^{2}.29^{2}.61.109  2^{7}.3  3.239
202  2^{8}.3^{4}.10273  2^{33}.3^{4}.17^{4}.17568767^{2}  2^{4}  2^{3}.2381
203  2^{35}.3^{7}.17.29.37.7547  2^{22}.5^{2}.7^{2}  2^{14}.3^{3}  2^{3}.193
204  2^{2}  2^{102}.3^{23}.7^{2}.11^{2}.13.17^{3}  2  2^{2}.3.5^{4}
205  2^{30}.3^{2}.5.13.229^{2}  −2^{14}.13^{2}.31^{2}.37^{2}.239  2^{15}.3  7^{2}.23
206  2^{16}.3^{2}.5^{2}.13.29.359  2^{12}.3^{2}.5^{2}.17^{4}.19^{2}.67^{2} …  2^{6}.3.5  17.2843
… 1801.411721^{2}
207  2^{20}.3^{4}.5^{2}.11^{2}  2^{28}.3^{8}.5^{5}.7^{2}.11^{4}  2^{7}.3^{2}.11  2^{6}
208  2^{14}.3^{2}.17  0  2^{7}.3  ?
209  2^{32}.3^{4}.15427.2002061  2^{10}.3^{4}.5^{5}.7.19  2^{9}  3.11^{2}
210  2^{8}  2^{176}.3^{14}.5^{8}.7^{7}.11^{2}.13^{2}.17^{4}  2^{4}  2^{3}.3
211 2^{14}.3.5.7^{4}.41^{2}.43.229.52184516509  1  2^{6}.7.41  2^{5}.5.23
212  2^{4}.3^{3}.7^{3}  2^{56}.3^{12}.5^{4}.7.37^{3}.151  2.7  2^{2}.13^{3}
213  2^{20}.3^{2}.5^{4}.13.89  3^{14}.5^{2}.19^{2}.37.61^{2}.229.257^{4}  2^{10}.3  2^{3}.3.17.53
214  2^{16}.3^{6}  −2^{24}.5^{4}.7^{4}.11^{2}.41.109^{2}.521 …  2^{6}.3^{2}  3.67.4903
… 1667^{2}.19079^{2}
215  2^{14}.3.7^{4}.101.107.321821.1933097  −2^{22}.5^{2}.31^{2}.41  2^{7}.7^{2}  2^{2}.3^{6}
216  2^{8}.3^{4}  0  2^{4}.3^{2}  ∞
217  2^{31}.3^{8}.11.31.557.619  5^{2}.19^{2}.31^{2}.281  2^{14}  2^{4}.5.43
218  2^{19}.3^{4}.5.11^{2}.23  2^{34}.3^{4}.7^{7}.41^{2}.167.601.7537^{2}  2^{7}.11  2.7.21227
219  2^{24}.3^{2}.29.73.1189637  −2^{16}.3^{8}.5^{4}.13^{2}.17^{2}.23.29^{2}.61  2^{10}.3  3^{2}.41.71
220  2^{2}  2^{77}.3^{16}.5^{4}.7^{11}.11^{4}.17.19^{3}  2  2.3.5
221  2^{40}.3^{3}.5^{6}.7.37.109^{2}.229  −2^{12}.3^{3}  2^{17}.3.5^{2}  5.107
222  2^{8}.3^{2}  −2^{60}.3^{14}.5^{10}.7^{4}.11^{4}.13^{5}.19^{2} …  2^{4}.3  2^{4}.3^{5}.11^{2}

… 23^{2}.31.37^{2}.109.389^{2}.409

223 2^{18}.7^{2}.19.103.3995922697473293141  1  2^{6}.7  2.3^{2}.5.641
224  2^{18}.5^{2}  0  2^{9}  ∞

N  Δ^{new}  Δ/Δ^{new}  [T̃^{new} : T^{new}] [T : W]
225  2^{12}.3^{2}.5^{5}  2^{38}.3^{10}  2^{6}.3.5^{2}  ∞
226  2^{23}.3.5^{3}  −2^{34}.3^{12}.7^{9}.11^{4}.23.41^{2} …  2^{7}  2^{2}.190129
… 107^{2}.167
227 2^{19}.5^{3}.7^{4}.13^{2}.29.31^{2} … 1 2^{7}.5.7.13.31 2^{2}.29^{2}
… 13591.57139.273349
228  2^{4}.3.11  −2^{88}.3^{36}.5^{12}.7.11^{4}  2^{2}  2^{2}.3^{9}
229  2^{24}.107.17467.39555937 × …  1  2^{9}  2^{2}.11.193
… 53625889
230 2^{16}.3^{2}.5.7.13.367 2^{52}.3^{2}.5^{18}.11^{4}.17^{4}.19^{2} … 2^{8} 2^{2}.5^{2}.53

… 41^{2}.43^{2}.47.53^{2}

231  2^{20}.3^{4}.5^{3}.7.31.229  −2^{59}.3^{16}.5^{12}.7^{3}.11^{3}  2^{10}.5  2^{2}.29
232  2^{20}.71  0  2^{7}  ?
233  2^{22}.3^{7}.53.139.653.4127 × …  1  2^{7}.3^{3}  2^{2}.3.11.197
… 24989.8388019
234  2^{6}.3^{2}.5^{2}  −2^{118}.3^{13}.5^{6}.7^{9}.11.17^{3}  2^{3}.3.5  2^{5}.3^{2}
235  2^{18}.3^{2}.73.1453.3851.1916279  2^{24}.3^{8}.19^{3}.103^{2}  2^{9}.3  2^{2}.3.17.31
236  2^{2}.3.7^{2}.107  −2^{55}.3^{12}.7.19^{4}.31^{3}.557^{3}  2.7  223^{2}
237  2^{18}.19.103.1705391537  2^{28}.3^{2}.5^{2}.11^{2}.31^{2}.83^{2}.983^{2}  2^{6}  2^{6}.5.97
238  2^{16}.5  −2^{73}.3^{11}.5^{2}.7^{4}.61^{2}.71^{2}.131^{2} …  2^{8}  2^{2}.13.331

… 311^{2}.337.479.1459^{2}

239 2^{6}.7^{2}.2833.51817.97423 × … 1 2^{3} 2^{2}.3.11^{2}
… 1174779433.8920940047
240  2^{10}  0  2^{5}  ∞
241  2^{23}.97.1489.20857 × …  1  2^{7}  2^{7}.151
… 651474368435017
242  2^{18}.3^{2}.5^{2}.11^{8}  2^{41}.3^{12}.5^{4}.7^{2}  2^{7}.11^{4}  ∞
243  2^{13}.3^{40}  0  2^{4}.3^{15}  ∞
244  2^{4}.5077  2^{60}.3^{10}.7.13^{6}.37^{3}.151.229^{2}  2  2^{2}.3.5.83^{2}
245  2^{36}.3^{4}.7^{8}.17  −2^{28}.3^{8}.5.17^{2}  2^{12}.3^{2}.7^{4}  2^{2}
246  2^{12}.3^{2}.5^{2}  2^{96}.3^{8}.5^{10}.7^{6}.11^{2}.23^{7}.37^{4} …  2^{6}.3.5  7.73^{2}
… 79^{2}.191^{2}
247 2^{30}.3^{4}.5^{2}.11.31^{2}.619 × … −2^{6}.3^{4}.5^{2} 2^{15}.31 2^{2}.29.241
… 57713.2655049
248  2^{22}.3.11.79  0  2^{10}  ?
249  2^{33}.389.23029  −2^{8}.3^{2}.5^{2}.11.31^{2}.197^{2}.2711^{2} …  2^{11}  2.13.41.89

… 11497^{2}.295201

250  2^{8}.5^{12}  0  2^{4}.5^{4}  ∞
251  2^{14}.5^{2}.29.373 × …  1  2^{4}  2^{2}.3.797
8768135668531.2006012696666681
252  2^{2}  2^{136}.3^{30}.5^{2}.7^{3}  2  ∞
253  2^{28}.3^{4}.13^{2}.2711.3187.170701  −2^{12}.3^{2}.5^{7}.7.13.19^{2}.23^{2}  2^{14}  2.4507
254  2^{28}.3^{4}.17.569  2^{16}.3^{18}.7^{2}.17^{2}.41.71 × …  2^{11}  7.189713

… 383^{2}.1231.86235899^{2}

255  2^{23}.5.13.229.1721  −2^{99}.3^{14}.5^{2}.11^{3}.13^{3}.17^{4}  2^{10}  2^{3}.3^{3}.379
256  2^{31}  0  2^{14}  ∞
257  2^{29}.29.479.71711.409177 × …  1  2^{11}  2.29.3251
… 654233.32354821
258 2^{12}.3^{2}.5^{2} −2^{104}.3^{14}.5^{16}.7^{17} × … 2^{6}.3.5 7^{3}.199^{2}

… 11^{2}.17.19^{2}.37.71^{2}.109

259  2^{43}.3^{9}.5^{3}.7^{2}.17^{3}.29.37.167  2^{4}.3^{12}.7^{4}  2^{20}.3.17  2^{2}.3.29.193
260  2^{6}.3.47  −2^{117}.3^{26}.5^{6}.7^{5}.11^{2}.17.19^{2}  2^{2}  ∞
261  2^{19}.3^{4}.5^{3}.23^{2}.229  2^{47}.3^{8}.5^{2}.23^{4}.73.229^{2}  2^{8}.3^{2}.23  2^{10}.5

N  Δ^{new}  Δ/Δ^{new}  [T̃^{new} : T^{new}]  [T : W]
262  2^{19}.3^{3}.5.11^{2}.13  −2^{43}.3^{4}.5^{2}.11^{2}.71 × …  2^{7}.3.11  29.482689
… 313^{2}.46141^{2}.75619573^{2}
263 2^{10}.11.61.397.15631853 × … 1 2^{5} 2^{3}.19.223

… 34867513.97092067.252746489

264  2^{6}.3^{2}  0  2^{3}.3  ∞
265  2^{37}.3^{2}.5^{4}.7^{3}.11^{2}.13.17^{2}.113  2^{26}.3^{4}.5^{3}.31^{2}.37^{2}.239  2^{13}.5.7.11.17  2.3^{3}.103
266  2^{16}.5.7.13.29.67  2^{51}.3^{31}.5^{6}.7^{6}.11^{6}.13^{4}.17.19^{3}.41.47.229^{2}  2^{8}  3.216679
267  2^{26}.3^{4}.5^{2}.7^{4}.13^{2}.97.241  −2^{17}.3^{4}.5^{8}.11.13^{2}.17^{2}.37.113^{2}.911.6689^{2}  2^{13}.5.7  3^{2}.17.41.271
268  2^{4}.3^{3}.5.7  −2^{46}.3^{20}.5^{20}.11^{2}.19^{4}.41.43^{2}  2^{2}.3  29^{2}.31^{2}
269  2^{22}.3^{2}.43.151.27767.65657.5550873754172978311  1  2^{6}.3  11^{2}.19^{2}
270  2^{8}  0  2^{4}  ∞
271 2^{12}.3^{2}.1367.6091.592661.1132673.14171513.172450541  1  2^{6}.3  2.3.17239
272  2^{28}.3^{3}.5  0  2^{13}.3  ?
273  2^{25}.79.4357  −2^{111}.3^{9}.5^{4}.7^{6}.11.79^{2}  2^{9}  2^{2}.3.313
274  2^{20}.3^{2}.19.37.1321  −2^{26}.5^{4}.7^{2}.11^{2}.29^{2}.109^{2}.149^{2}.401^{2}.769.47143.895241^{2}  2^{7}.3  2.11.304937
275  2^{31}.3^{8}.5^{12}.7^{2}.11^{2}.13^{2}  −2^{28}.3^{6}.5^{4}.7^{4}.19  2^{12}.3^{3}.5^{5}.7  2^{2}
276  2^{10}.5  −2^{108}.3^{19}.5^{21}.7^{9}.11^{11}.41^{2}  2^{2}  2^{4}.7^{4}
277  2^{22}.5^{2}.19.29.37.137^{2}.92767.1530091.25531570859  1  2^{10}.5.137  2.3^{2}.8311
278  2^{22}.3^{4}.7.17^{2}.103.107  −2^{38}.3^{4}.7^{7}.41^{2}.167.271^{3}.997^{2}.2151701^{2}  2^{8}.17  6084451
279  2^{28}.3^{5}.5^{2}.229.1373^{2}  2^{60}.3^{8}.5^{5}.229^{2}  2^{11}.3^{2}  2^{9}.5^{3}
280  2^{18}.3.11.17  0  2^{9}  ∞
281  2^{22}.3.5.181.857.8388019.2647382149.1778899342669  1  2^{7}  2^{2}.79309
282  2^{23}.3^{2}.37  2^{96}.3^{13}.5^{4}.7^{6}.17^{2}.19^{4}.23^{2}.43^{4}.47^{4}.103^{4}.457^{2}.3529^{2}  2^{8}  2^{7}.5^{6}
283  2^{40}.349.1297.413713.73199099.5832488839  1  2^{9}  2^{7}.2687
284  2^{6}.3^{5}.107  2^{52}.3^{52}.7.103.257^{6}  2^{3}  3^{2}.67^{2}
285  2^{34}.3^{3}.7  −2^{80}.3^{23}.5^{5}.7^{2}.11^{3}.19^{3}.23.37^{2}.47^{2}.73.709^{2}  2^{12}.3  2.3.10771
286  2^{16}.3^{2}.31^{2}  −2^{70}.3^{22}.5^{9}.7^{4}.13^{2}.19^{4}.103^{2}.139^{2}.257.5560463^{2}  2^{8}.3  120671
287  2^{28}.3.5^{2}.7^{2}.61^{2}.97^{2}.103.257.211039.1798619  −2^{8}.13^{2}.19^{2}.37^{4}.1847  2^{14}.61.97  2^{2}.3.823
288  2^{20}.3^{2}  0  2^{10}.3  ∞
289  2^{23}.3^{16}.13^{2}.17^{8}  2^{6}.3^{4}  2^{6}.3^{4}.17^{4}  ∞
290  2^{20}.3^{3}.7.13^{2}.23.67  −2^{84}.3^{6}.5^{10}.7^{11}.11.13^{2}.17^{3}.19^{5}.23^{2}.31.37^{4}.151  2^{10}  2^{3}.5.97.421
291  2^{32}.3^{4}.5^{4}.11^{2}.13.19.457.16657  −2^{22}.7^{4}.11^{2}.13^{2}.23^{2}.139^{2}.491.1933.2777^{2}  2^{15}.3^{2}.5.11  73.167
292  2^{7}.5.41^{2}  −2^{64}.3^{28}.5^{3}.7.13^{3}.17.19^{4}.101^{2}.389^{2}  2^{2}  2.3^{2}.97^{2}
293  2^{26}.3^{2}.29.233.2351^{2}.69763.42711913589792108923  1  2^{10}  67759
294  2^{12}.3^{2}.7^{4}  2^{147}.3^{16}.5^{6}.7^{21}.17^{2}  2^{6}.3.7^{2}  ∞
295  2^{36}.3^{4}.7^{4}.43.37199.8055869  −2^{6}.7^{2}.31^{2}.107^{2}.557^{2}.947^{2}.271499  2^{15}  2.223.241
296  2^{18}.11.53.83.229  0  2^{9}  ?
297  2^{30}.3^{24}.47^{2}  0  2^{11}.3^{10}  ∞
298  2^{21}.3.5^{2}.7.13^{2}.103.107  2^{36}.3^{6}.7^{4}.13^{2}.23^{2}.29^{2}.41.239.1847.234893^{2}.1252037^{2}  2^{8}.5  2^{3}.2972969
299  2^{50}.3.5^{9}.7.17^{3}.197.5936311524617  5^{2}.11^{2}.17^{4}.43^{2}  2^{19}.5^{3}.17  2^{3}.3769
300  2^{4}.5^{4}  2^{113}.3^{42}.5^{16}.7^{3}.11^{4}  2^{2}.5^{2}  ∞
301  2^{38}.19.31.37.103.9739.81509.32366197  −2^{30}.7^{3}.17^{2}.113  2^{19}  2^{2}.3.13.17.61
302  2^{33}.3^{4}.389.1613  2^{18}.3^{8}.5^{4}.7^{7}.11^{2}.41^{2}.67^{4}.167.257^{2}.431^{2}.16937.439867^{2}  2^{11}.3^{2}  17.877.4457
303  2^{35}.89.3083683.3678833  2^{39}.3^{2}.7^{2}.59.1103^{2}.4021.17568767^{2}  2^{15}  2^{4}.61.199
304  2^{26}.3^{4}.31^{2}  0  2^{13}.3^{2}  ?
305  2^{52}.3^{4}.31.43.977.2777.82219057  2^{8}.3^{6}.11.37^{4}.139^{2}.2371  2^{19}  2^{2}.3^{4}.223
306  2^{24}.3^{4}  2^{183}.3^{30}.5^{4}.7^{4}.17^{6}  2^{9}.3  3^{5}.5^{4}
307 2^{22}.3^{2}.5^{5}.11^{2}.13^{3}.107^{2}.457.3697.21577.974513.568380457  1  2^{11}.3.5^{2}.11.13.107 2^{3}.3^{2}.16529
308  2^{12}.3.127  −2^{140}.3^{38}.5^{17}.7.11^{2}  2^{3}  2^{9}.23
309  2^{24}.5.37.81509.109363884517  2^{32}.3^{4}.5^{8}.11^{2}.17^{2}.89^{2}.397.411721^{2}  2^{10}  2^{6}.3.11^{2}.421
310  2^{23}.3^{2}.37  2^{119}.3^{10}.5^{12}.7^{9}.11^{4}.17.19^{4}.23^{2}.29^{2}.41^{2}.73^{2}.5077^{2}  2^{8}  2.5^{2}.7.73.991

N  Δ^{new}  Δ/Δ^{new}  [T̃^{new} : T^{new}]  [T : W]
311  2^{8}.5^{2}.29.3013091897.2106873009119126062143259000543887593  1  2^{4}  2.610843
312  2^{18}  0  2^{9}  ∞
313  2^{24}.5.41^{2}.8619587.9614923.130838023.2164322751511  1  2^{11}.41  2^{5}.19.241
314  2^{14}.3.47.53.337.907.176130259  2^{30}.5^{2}.7.19^{2}.61^{2}.113^{2}.397^{2}.709^{2}.743.1489.48795779^{2}  2^{7}  2.13^{2}.127.461
315  2^{24}.3^{4}.5^{3}.17  −2^{158}.3^{16}.5^{6}.7^{3}.11.13.17^{3}  2^{9}.3^{2}.5  2^{3}.3
316  2^{6}.3^{4}.13^{2}  2^{92}.3^{18}.5^{4}.7.17.53^{4}.83^{3}.271.983^{3}  2^{3}.3^{2}  2^{4}.541^{2}
317  2^{26}.7.367.3217.660603043.14989400036918065702697531  1  2^{11}  5.167.14627
318  2^{16}.5^{2}.17.41  −2^{84}.3^{16}.5^{8}.7^{7}.11^{4}.13^{2}.17^{2}.19^{2}.37^{4}.43^{2}.71.103^{2}.107^{4}.151^{2}.227^{2}.263.1054013^{2}  2^{8}.5  2^{4}.3^{2}.5^{4}
319  2^{34}.3^{4}.23^{2}.127.1063.1433.2459.2777.6709.65551  −2^{12}.3^{4}.5^{2}.7.17^{2}.29.167^{2}.239  2^{17}.23  2.13.59.197
320  2^{39}  0  2^{18}  ∞
321  2^{32}.5^{2}.21803.24443.826957  −2^{12}.5^{4}.7^{2}.29^{2}.41^{2}.61.1667^{2}.14051^{2}.19079^{2}.1229279  2^{12}  2^{7}.11.73.109
322  2^{28}.3^{3}.5.79  −2^{100}.3^{5}.5^{18}.7^{3}.11^{4}.19^{4}.23^{2}.29^{2}.37^{2}.41^{3}.151.536777^{2}  2^{12}.3  2^{3}.13.41.53.67
323  2^{48}.7^{2}.17^{3}.19.73.103.353.1453.1627.1697.3851.17299  2^{12}.3^{5}.5^{3}.7^{2}.59  2^{23}.7.17  2^{2}.767603
324  2^{4}.3^{8}  0  2^{2}.3^{4}  ∞
325  2^{49}.3^{9}.5^{12}.7^{4}.37^{2}  2^{50}.3^{10}  2^{17}.3^{4}.5^{6}.7^{2}  2^{2}.3.17
326  2^{18}.17^{2}.617.28921.482689  −2^{35}.3^{16}.5^{2}.13^{2}.47^{2}.61^{2}.263.14831.65657^{2}.82536739^{2}  2^{9}.17  311.102305897
327  2^{37}.5.37.139.1023203.1033895651  2^{40}.3.7^{4}.13^{2}.107^{2}.139^{2}.491.577.7537^{2}  2^{11}  2^{6}.3.281.463
328  2^{28}.3^{3}.37.197  0  2^{11}.3  ?
329  2^{46}.5^{3}.7^{4}.11.13^{2}.17.37^{3}.1447.1609.51607  3^{6}.19^{4}.31^{2}.83^{2}.103^{2}.18617  2^{20}.5.13.37  2^{4}.11.1801
330  2^{8}  2^{264}.3^{26}.5^{21}.7^{22}.11^{10}.17^{3}.19^{4}.23.37^{2}  2^{4}  2.3.5.1153
331  2^{32}.3^{2}.53^{2}.229.1399.21911.205493.6363601.584461573862449  1  2^{12}.3.53  2^{2}.1120529
332  2^{9}.7^{3}.29^{2}  −2^{82}.3^{12}.5^{2}.7.71.131^{4}.197^{3}.229^{2}.11497^{3}  2^{2}.29  2^{2}.5^{2}.479^{2}
333  2^{42}.3^{7}.5^{2}.7^{2}.37.389  2^{72}.3^{9}.5^{4}.7^{4}.11.37^{2}.389^{2}  2^{13}.3^{2}.5.7  2^{10}.3^{5}
334  2^{27}.5^{2}.7^{3}.67.733  2^{18}.3^{10}.5^{4}.7^{4}.11^{4}.41.113^{2}.8269^{2}.1951993.5103536431379173^{2}  2^{12}.7  113.3715823
335  2^{31}.3.5.29.71.83.179^{2}.887.26393.6262079.23057641  −2^{22}.3^{12}.5^{8}.11^{2}.41.59^{2}  2^{13}.179  2^{2}.3.331.431
336  2^{16}.3^{2}  0  2^{8}.3  ∞
337  2^{28}.113.593.2791.2963615537.747945736667.4122851467451  1  2^{13}  2^{4}.3^{3}.27299
338  2^{16}.3^{2}.7^{6}.13^{8}  2^{36}.3^{8}.5^{2}.7^{14}.13^{8}.41^{4}.167^{2}  2^{8}.3.7.13^{4}  ∞
339  2^{41}.7^{2}.13^{2}.17.71^{2}.8297.470621  −2^{29}.3^{18}.7^{6}.11^{8}.13^{5}.107^{2}.167.647  2^{14}.7.13.71  2^{2}.5^{2}.127.11801
340  2^{6}.101  −2^{198}.3^{37}.5^{8}.7^{7}.17^{3}  2^{2}  ∞
341  2^{44}.3.5^{3}.89.151.1121599.344460847.14444130109  −2^{14}.3^{3}.5^{10}.7^{4}.13^{2}  2^{16}  31.9043
342  2^{12}.3^{4}.5^{2}  −2^{184}.3^{58}.5^{16}.7^{3}.11^{6}  2^{6}.3^{2}.5  2.3^{9}.5^{2}
343  2^{27}.7^{48}.13^{2}.29^{3}.41^{2}  0  3.33521863.4906673923  ∞
344  2^{26}.3.229.1999567  0  2^{11}  ?
345  2^{46}.3^{2}.5^{4}.79  −2^{100}.3^{13}.5^{19}.7^{4}.11^{13}.13^{2}.17^{4}.23.53^{2}  2^{18}.5^{2}  2^{2}.7^{2}.15991
346  2^{18}.7^{2}.229.2777.2075621  2^{40}.5^{4}.7^{6}.29^{2}.31^{2}.67^{2}.79^{2}.311.1279.1289.5608385124289^{2}  2^{9}.7  2^{3}.18947.256957
347 2^{27}.5.7^{2}.19^{2}.331.349.479.617.1797330450291217.918291275915301361  1  2^{10}.7.19  2^{6}.3.7.19709
348  2^{8}  2^{129}.3^{27}.5^{17}.7^{10}.11^{7}.13^{4}.17^{2}.23^{6}.41.47.73^{3}.229^{3}  2^{4}  2^{8}.5.11^{4}
349  2^{28}.13.103.1118857.72318613.6771977049413.1313981654817031  1  2^{11}  2^{3}.3^{2}.239.1531
350  2^{24}.3^{4}.5^{6}  2^{138}.3^{34}.5^{29}.11^{4}.17^{6}.41^{2}  2^{9}.3.5^{3}  2^{3}.3

N  Δ^{new}  Δ/Δ^{new}  [T̃^{new} : T^{new}]  [T : W]
351  2^{32}.3^{17}.5^{4}.13^{2}.19.37^{2}  0  2^{12}.3^{8}  ∞
352  2^{42}.17^{2}  0  2^{21}  ∞
353  2^{34}.3^{2}.5.127^{2}.229.114641.551801.12611821.7779730837.24314514437  1  2^{15}.3.127  2^{2}.3.47.26927
354  2^{28}.5^{2}.11.79  −2^{98}.3^{14}.5^{12}.7^{4}.11^{6}.19^{6}.29^{2}.31^{8}.41^{2}.127.229^{6}.557^{4}.20627^{2}  2^{12}.5  2^{15}.3^{4}.5
355  2^{51}.5^{2}.19.29.103.62581037.933591331  3^{10}.5.7^{2}.19.61^{2}.103^{2}.257^{4}.359^{2}.3671  2^{23}  2^{6}.3^{2}.5^{2}.13.37
356  2^{9}.3.4079.31547  −2^{114}.3^{14}.5^{9}.7^{6}.71^{2}.199.6689^{3}  2  2^{5}.11.487^{2}
357  2^{49}.3.79.113  −2^{99}.3^{16}.7^{4}.17^{4}.47^{2}.71^{2}.131^{2}.223^{2}.311^{2}.397.1459^{2}.1753  2^{18}  2^{2}.17.86341
358  2^{22}.3^{3}.5^{3}.7.11^{4}.17^{3}  −2^{46}.3^{8}.5^{2}.7^{4}.13^{2}.17^{4}.29^{2}.79^{2}.239.313^{2}.3257.137707^{2}.536747147^{2}  2^{11}.3.11^{2}  251.4099597
359  2^{24}.3^{4}.2777.16512254293.64542630435970307.2171776478013633068927  1  2^{11}  2^{6}.5.123059
360  2^{12}.3^{2}  0  2^{6}.3  ∞
361  2^{28}.3^{10}.5^{11}.11^{4}.19^{12}  2^{6}.3^{2}.5^{2}  2^{12}.3.5^{2}.11^{2}.19^{6}  ∞
362  2^{26}.5^{4}.17^{3}.6359.1134769  2^{40}.3^{2}.5^{4}.7^{4}.17^{2}.23^{2}.29^{2}.47.61^{2}.71.97.397^{2}.797^{2}.5297.595051637^{2}  2^{10}.5.17  2^{2}.2609601041
363  2^{36}.3^{9}.5^{7}.11^{18}  2^{38}.3^{22}.5^{8}.11^{4}  2^{15}.3^{3}.5.11^{8}  ∞
364  2^{15}.3^{2}  2^{125}.3^{63}.5^{10}.7^{12}.11^{4}.23.79^{3}  2^{5}  2^{7}.3^{2}.5.17.37
365  2^{51}.3^{3}.5^{4}.7^{3}.47.313^{2}.353783.9377894173  −2^{20}.3^{12}.5^{4}.11^{2}.13^{4}.19^{2}.29.269  2^{19}.3.5^{2}  3^{2}.37.71473
366  2^{16}.3^{2}.13^{2}.17  −2^{162}.3^{11}.5^{4}.7^{6}.13^{10}.17^{3}.19^{6}.23^{3}.31^{2}.37^{6}.127^{2}.137.151^{3}.229^{2}.5623^{2}  2^{8}.3.13  2^{4}.5^{7}.1277^{2}
367  2^{22}.7.81421.251387.418175501.15354151381.13144405392643360366681  1  2^{11}  2^{5}.3197809
368  2^{40}.3^{4}.5^{3}.17  0  2^{20}.3^{2}.5  ?
369  2^{41}.3^{6}.5^{2}.23^{2}.37^{3}.79  −2^{86}.3^{12}.5^{4}.23^{4}.37^{3}.79^{2}.191  2^{15}.3^{3}.5.23  7.13^{2}
370  2^{28}.3^{2}.11.223  −2^{148}.3^{28}.5^{10}.7.11^{2}.13^{4}.19^{2}.23.31^{2}.61.193.269.23029^{2}.60869^{2}  2^{12}  2^{4}.72709379
371  2^{62}.5^{3}.7.157.229.64627370911.76723322773093  2^{30}.3^{3}.5^{2}.37^{2}.191^{2}.643  2^{25}.5  2^{5}.3.5.76103
372  2^{10}.5^{2}.17  −2^{181}.3^{31}.5^{13}.7^{4}.11^{12}.13^{2}.17^{2}.19^{4}.41^{2}.127.229^{3}  2^{5}.5  2^{2}.3^{5}.5^{8}
373  2^{32}.7.11^{3}.23.199.673.2143.1542194372227.72819251148518000363297  1  2^{13}.11  2^{4}.3.1520821
374  2^{28}.5^{2}.157.257.11117.17417  2^{142}.3^{20}.5^{18}.7^{2}.11^{2}.13^{2}.17^{2}.23.37^{4}.8461^{2}  2^{14}  11.3259.109111
375  2^{24}.5^{24}.101^{2}  0  2^{12}.5^{8}  ∞
376  2^{33}.5^{5}.41^{2}.61  0  2^{15}  ?
377  2^{70}.3^{3}.5^{2}.7^{2}.829.991.36497.202817.400031.54126547  2^{6}.7^{2}.23^{3}.31^{2}.79  2^{30}.3.7  181.118057
378  2^{16}.3^{8}  0  2^{8}.3^{4}  ∞
379  2^{28}.59.317.421.278329.5698591.2117788336277.2851210737989187265253  1  2^{13}  2^{5}.3^{3}.7.229.809
380  2^{15}.3  2^{158}.3^{47}.5^{9}.7^{2}.11^{7}.13^{4}.17^{2}.19.37^{3}.113.151.709^{3}  2^{5}  2^{2}.3^{4}.11.53
381  2^{32}.11^{2}.15427.81509.98361184089911  2^{36}.3^{10}.5^{2}.7^{2}.19^{2}.71.109^{2}.647^{2}.2003.86235899^{2}  2^{12}.11  2^{6}.3.7^{3}.13.43.113
382  2^{28}.7^{2}.11.13^{2}.619.44171  2^{20}.3^{6}.5^{4}.11^{4}.41.967^{2}.1511.6277^{2}.63703.382146223^{2}.319500117632677^{2}  2^{13}  5.59.9181.50459
383 2^{16}.5.11^{2}.13.72893.3151861.16141144314299.178236551484825400362837637090811  1  2^{8}.11  2^{7}.2480227
384  2^{40}  0  2^{20}  ∞
385  2^{61}.3.37^{3}.2837  −2^{114}.3^{18}.5^{17}.7^{8}.11.17^{2}.19^{5}.389  2^{24}  2^{4}.3.5.29.71
386  2^{32}.5^{2}.14653.22961.9659021  −2^{30}.3^{2}.5^{4}.7.11^{4}.17^{2}.19^{2}.29^{2}.31^{2}.103^{2}.229^{2}.271^{2}.337.401^{2}.4153^{2}.244553.680059^{2}  2^{12}  3^{2}.13^{2}.74891419
387  2^{51}.3^{9}.5^{8}.7^{2}.13^{2}.71  −2^{106}.3^{16}.5^{6}.7^{4}.71^{2}  2^{20}.3^{4}.5^{3}.7  2.7^{3}.17^{2}
388  2^{8}.7^{2}.179.1297  −2^{90}.3^{16}.7^{11}.67^{4}.71^{4}.137^{2}.223^{2}.433.2777^{3}  2^{3}  2^{3}.3^{3}.22283^{2}
389  2^{53}.3^{4}.5^{6}.31^{2}.37.389.3881.215517113148241.477439237737571441  1  2^{17}.3^{2}.5.31  7^{2}.67.173.863