Ima­gin­ary tri­quad­rat­ic num­ber fields with ex­po­nent 1,3,5.

The scientific content is given in the article:

J. Klüners, T. Komatsu, Imaginary multiquadratic number fields of exponent 3 and 5

Ex­po­nent 1

The following table contains all imaginary biquadratic fields of family 3a with exponent 1.

The results are proven without using any conjecture.

Discriminant Factorization Generators Class group
5308416 2^16 * 3^4 -3, -4, -8 C1
49787136 2^8 * 3^4 * 7^4 -3, -4, -7 C1
303595776 2^8 * 3^4 * 11^4 -3, -4, -11 C1
796594176 2^12 * 3^4 * 7^4 -3, -7, -8 C1
959512576 2^16 * 11^4 -4, -8, -11 C1
2702336256 2^8 * 3^4 * 19^4 -3, -4, -19 C1
80102584576 2^8 * 7^4 * 19^4 -4, -7, -19 C1
154550410641 3^4 * 11^4 * 19^4 -3, -11, -19 C1

 

The following table contains all imaginary biquadratic fields of family 3b with exponent 1. The results are proven without using any conjecture.

Discriminant Factorization Generators Class group
12960000 2^8 * 3^4 * 5^4 -3, -4, 5 C1
40960000 2^16 * 5^4 -4, -8, 5 C1
121550625 3^4 * 5^4 * 7^4 -3, -7, 5 C1
207360000 2^12 * 3^4 * 5^4 -3, -8, 5 C1
384160000 2^8 * 5^4 * 7^4 -4, -7, 5 C1
4857532416 2^12 * 3^4 * 11^4 -3, -11, 8 C1
6146560000 2^12 * 5^4 * 7^4 -7, -8, 5 C1
17555190016 2^8 * 7^4 * 13^4 -4, -7, 13 C1
99049307841 3^4 * 11^4 * 17^4 -3, -11, 17 C1

Ex­po­nent 3

The following table contains all imaginary triquadratic fields of family 3a with exponent 3. The results are proven using ERH for imaginary quadratic number fields.

Discriminant Factorization Generators Class group
2847396321 3^4 * 7^4 * 11^4 -3, -7, -11 C3
8540717056 2^16 * 19^4 -4, -8, -19 C3
19150131456 2^8 * 3^4 * 31^4 -3, -4, -31 C3
54423757521 3^4 * 7^4 * 23^4 -3, -7, -23 C3
70892257536 2^8 * 3^4 * 43^4 -3, -4, -43 C3
251265597696 2^8 * 3^4 * 59^4 -3, -4, -59 C3
306402103296 2^12 * 3^4 * 31^4 -3, -8, -31 C3 x C3
417853645056 2^8 * 3^4 * 67^4 -3, -4, -67 C3
794123370496 2^16 * 59^4 -4, -8, -59 C3 x C3
984095744256 2^8 * 3^4 * 83^4 -3, -4, -83 C3 x C3
1048870932736 2^8 * 11^4 * 23^4 -4, -11, -23 C3
4054427900721 3^4 * 11^4 * 43^4 -3, -11, -43 C3 x C3
7740770386176 2^8 * 3^4 * 139^4 -3, -4, -139 C3 x C3
8590432731136 2^16 * 107^4 -4, -8, -107 C3 x C3
12813994352896 2^8 * 11^4 * 43^4 -4, -11, -43 C3 x C3
32464571577361 7^4 * 11^4 * 31^4 -7, -11, -31 C3 x C3
55383125856256 2^12 * 11^4 * 31^4 -8, -11, -31 C3 x C3
119168138285056 2^12 * 7^4 * 59^4 -7, -8, -59 C3 x C3
177878343106816 2^8 * 11^4 * 83^4 -4, -11, -83 C3 x C3
229451724656896 2^8 * 7^4 * 139^4 -4, -7, -139 C3 x C3
248906913382656 2^8 * 3^4 * 331^4 -3, -4, -331 C3 x C3 x C3
442705543843761 3^4 * 11^4 * 139^4 -3, -11, -139 C3 x C3
466728668041216 2^12 * 7^4 * 83^4 -7, -8, -83 C3 x C3
491298928189696 2^8 * 11^4 * 107^4 -4, -11, -107 C3 x C3
2350637069590161 3^4 * 11^4 * 211^4 -3, -11, -211 C3 x C3
3918727948865536 2^12 * 23^4 * 43^4 -8, -23, -43 C3 x C3
13142294978742801 3^4 * 43^4 * 83^4 -3, -43, -83 C3 x C3 x C3

 

The following table contains all imaginary triquadratic fields of family 3b with exponent 5. The results are proven using ERH for imaginary quadratic number fields.

Discriminant Factorization Generators Class group
1871773696 2^16 * 13^4 -4, -8, 13 C3
14166950625 3^4 * 5^4 * 23^4 -3, -23, 5 C3
14666178816 2^8 * 3^4 * 29^4 -3, -4, 29 C3 x C3
43237380096 2^12 * 3^4 * 19^4 -3, -19, 8 C3
44774560000 2^8 * 5^4 * 23^4 -4, -23, 5 C3
46352367616 2^16 * 29^4 -4, -8, 29 C3
234658861056 2^12 * 3^4 * 29^4 -3, -8, 29 C3
280883040256 2^12 * 7^4 * 13^4 -7, -8, 13 C3
419936400625 5^4 * 7^4 * 23^4 -7, -23, 5 C3
517110562816 2^16 * 53^4 -4, -8, 53 C3 x C3
716392960000 2^12 * 5^4 * 23^4 -8, -23, 5 C3
952857108736 2^8 * 13^4 * 19^4 -4, -19, 13 C3
3351129310881 3^4 * 11^4 * 41^4 -3, -11, 41 C3
4020249563136 2^12 * 3^4 * 59^4 -3, -59, 8 C3 x C3
6752430919936 2^8 * 13^4 * 31^4 -4, -31, 13 C3
7815289901056 2^12 * 11^4 * 19^4 -11, -19, 8 C3
8510429245696 2^8 * 7^4 * 61^4 -4, -7, 61 C3 x C3
8936757492481 7^4 * 13^4 * 19^4 -7, -19, 13 C3
15245713739776 2^12 * 13^4 * 19^4 -8, -19, 13 C3 x C3
43489065701376 2^12 * 3^4 * 107^4 -3, -107, 8 C3 x C3 x C3
63330416557681 7^4 * 13^4 * 31^4 -7, -31, 13 C3
108038894718976 2^12 * 13^4 * 31^4 -8, -31, 13 C3 x C3
373448513433856 2^8 * 7^4 * 157^4 -4, -7, 157 C3 x C3
443091722465536 2^8 * 31^4 * 37^4 -4, -31, 37 C3 x C3
565268503941376 2^8 * 23^4 * 53^4 -4, -23, 53 C3 x C3 x C3
726672516714496 2^12 * 11^4 * 59^4 -11, -59, 8 C3 x C3
1023381597392896 2^12 * 7^4 * 101^4 -7, -8, 101 C3 x C3
3437435741863201 13^4 * 19^4 * 31^4 -19, -31, 13 C3 x C3
6468184485400576 2^12 * 19^4 * 59^4 -19, -59, 8 C3 x C3 x C3
7860782851035136 2^12 * 11^4 * 107^4 -11, -107, 8 C3 x C3
9044296063062016 2^12 * 23^4 * 53^4 -8, -23, 53 C3 x C3 x C3
69969611497148416 2^12 * 19^4 * 107^4 -19, -107, 8 C3 x C3 x C3
6505835909336928256 2^12 * 59^4 * 107^4 -59, -107, 8 C3 x C3 x C3 x C3

Ex­po­nent 5

The following table contains all imaginary triquadratic fields of family 3a with exponent 5. The results are proven using ERH for imaginary quadratic number fields.

Discriminant Factorization Generators Class group
224054542336 2^16 * 43^4 -4, -8, -43 C5
488455618816 2^8 * 11^4 * 19^4 -4, -11, -19 C5
1281641353216 2^12 * 7^4 * 19^4 -7, -8, -19 C5
5394359275776 2^8 * 3^4 * 127^4 -3, -4, -127 C5 x C5
12922702073856 2^12 * 3^4 * 79^4 -3, -8, -79 C5 x C5
14637786276096 2^8 * 3^4 * 163^4 -3, -4, -163 C5
55059011870976 2^8 * 3^4 * 227^4 -3, -4, -227 C5 x C5
75528336015616 2^8 * 11^4 * 67^4 -4, -11, -67 C5
181016143442176 2^8 * 7^4 * 131^4 -4, -7, -131 C5 x C5
672285245399296 2^8 * 19^4 * 67^4 -4, -19, -67 C5
950162376687616 2^16 * 347^4 -4, -8, -347 C5 x C5
2604748421533696 2^12 * 19^4 * 47^4 -8, -19, -47 C5 x C5
3044300929433856 2^8 * 3^4 * 619^4 -3, -4, -619 C5 x C5
3108741460575921 3^4 * 19^4 * 131^4 -3, -19, -131 C5 x C5
3847891608453376 2^8 * 11^4 * 179^4 -4, -11, -179 C5 x C5
4727561352765696 2^8 * 3^4 * 691^4 -3, -4, -691 C5 x C5
12187508427908401 7^4 * 19^4 * 79^4 -7, -19, -79 C5 x C5
26112925926363136 2^12 * 7^4 * 227^4 -7, -8, -227 C5 x C5
54341122488606976 2^8 * 11^4 * 347^4 -4, -11, -347 C5 x C5
102823251817101201 3^4 * 47^4 * 127^4 -3, -47, -127 C5 x C5 x C5
923878543897348081 7^4 * 43^4 * 103^4 -7, -43, -103 C5 x C5

 

The following table contains all imaginary triquadratic fields of family 3b with exponent 5. The results are proven using ERH for imaginary quadratic number fields.

Discriminant Factorization Generators Class group
16243247601 3^4 * 7^4 * 17^4 -3, -7, 17 C5
107049369856 2^8 * 11^4 * 13^4 -4, -11, 13 C5
122825015296 2^16 * 37^4 -4, -8, 37 C5
247033850625 3^4 * 5^4 * 47^4 -3, -47, 5 C5
1004006004001 7^4 * 11^4 * 13^4 -7, -11, 13 C5
1134276120576 2^12 * 3^4 * 43^4 -3, -43, 8 C5
2936017137361 7^4 * 11^4 * 17^4 -7, -11, 17 C5
7322571300625 5^4 * 7^4 * 47^4 -7, -47, 5 C5
12491983360000 2^12 * 5^4 * 47^4 -8, -47, 5 C5
13169822450625 3^4 * 5^4 * 127^4 -3, -127, 5 C5 x C5
29828735941761 3^4 * 19^4 * 41^4 -3, -19, 41 C5
35678353674496 2^8 * 13^4 * 47^4 -4, -47, 13 C5 x C5
41623142560000 2^8 * 5^4 * 127^4 -4, -127, 5 C5 x C5
42415313391616 2^12 * 11^4 * 29^4 -8, -11, 29 C5
98706291490816 2^16 * 197^4 -4, -8, 197 C5 x C5
136166867931136 2^12 * 7^4 * 61^4 -7, -8, 61 C5
334623934267441 7^4 * 13^4 * 47^4 -7, -47, 13 C5 x C5
390379551900625 5^4 * 7^4 * 127^4 -7, -127, 5 C5 x C5
2040495219329281 11^4 * 13^4 * 47^4 -11, -47, 13 C5 x C5 x C5
4810197031981056 2^12 * 3^4 * 347^4 -3, -347, 8 C5 x C5
17963217697171041 3^4 * 17^4 * 227^4 -3, -227, 17 C5 x C5
124309644679881441 3^4 * 11^4 * 569^4 -3, -11, 569 C5 x C5
607748470482582081 3^4 * 41^4 * 227^4 -3, -227, 41 C5 x C5
793389288712200625 5^4 * 47^4 * 127^4 -47, -127, 5 C5 x C5 x C5
864003980025204736 2^12 * 37^4 * 103^4 -8, -103, 37 C5 x C5 x C5
869457959817711616 2^12 * 11^4 * 347^4 -11, -347, 8 C5 x C5
2693876092569442561 11^4 * 29^4 * 127^4 -11, -127, 29 C5 x C5 x C5 x C5
3246907040793595201 11^4 * 17^4 * 227^4 -11, -227, 17 C5 x C5
203026005223874891776 2^12 * 43^4 * 347^4 -43, -347, 8 C5 x C5 x C5
977807264466179992321 19^4 * 41^4 * 227^4 -19, -227, 41 C5 x C5 x C5
Further information:
C-Programs

The following two c-programs can be used to compute all imaginary quadratic number fields with exponent less or equal to 8 and discriminant bound 3.1·1020.

Smallest Split fixed.c 

No small split.c

List of fields

Julia Programs

The following file contains the Julia code. You need to install Julia including the Hecke and the Markdown package. The function M1 computes the imginary quadratic number fields of given exponent. The functions M2a, M2b, M3a, and M3b compute the corresponding families.

Download File