Old Sextic Fields Ramifying Above {7, 19}

Discriminant class: −19

∞  7 19 a_{1} a_{2} a_{3} Over p = 7  Over p = 19  2 3 5 11 13 17 23 29
G  h  a_{4} a_{5} a_{6} f^{e} c_{w} d s f^{e} c_{w} d  s 31 37 41 43 47 53 59 61
1.  3_{i}  2_{+} 3_{k}  1 1  0 1^{3} 2_{+} 1  3^{2} 3_{k} 19∗  6 6 3 3 222 33 3 222
S_{3} × C_{3}  5 -4  1 3 0_{+} 1  6 6 222 33 3 6 6 3
2.  3_{i}  4_{+} 3_{k} -3 -4 13 1^{3} 2_{+} 1  1^{2} 1_{i} 19∗ 19∗ 222 222 33 33 222  33 222
S_{3}  3  7 -14 28 1^{3} 2_{+} 1  1^{2} 1_{i} 19∗ 19∗ 222 222 222 33 33 222 222 33
3.  3_{i}  4_{+} 3_{k} -1 10  -7 1^{3} 2_{+} 1  3^{2} 3_{k} 19∗  6 6 33 33 222 33 33 222
C_{6}  3  75 -16 239 1^{3} 2_{+} 1  6 6 222  33 6 6 33
4.  3_{i}  2_{+} 5_{k} -2 8 -12 1^{3} 2_{+} 1  1^{6} 5_{k} 19∗  222 6 33 3 6 33 3 6
S_{3} × C_{3} 3  22 -84 239 3 0_{+} 1  6 6 6 3 3 222 6 3
5.  1_{k}  2_{−} 5_{k}  2 8 -26 1^{2} 1_{k} 7 7 1^{6} 5_{k} 19∗  6 6 33 22 6 33 33 6
A_{4} × C_{2}  -16 27 11 1^{2} 1_{k} 7 7  2 2 6 33 33 6 6 33
6.  3_{i}  2_{+} 5_{k} -3 -1 -12 1^{3} 2_{+} 1  1^{6} 5_{k} 19∗  6 222 3 3 6  33 6
S_{3} × C_{3} 3  71 96 163 3 0_{+} 1  6 6 6 3 33 6 222 33
7.  3_{i}  4_{+} 5_{k} -1 2 -11 1^{3} 2_{+} 1  1^{6} 5_{k} 19∗  6 6 3 33 6 33 3 6
S_{3} × C_{3} 27  75 90 349 1^{3} 2_{+} 1  222 222 6 3 3 6 6 3
8.  3_{i}  4_{+} 5_{k}  1 2 11 1^{3} 2_{+} 1  1^{6} 5_{k} 19∗  222 6  33 6 33 33 6
C_{6}  9  341 708 2876 1^{3} 2_{+} 1  6 6 6 33 33 222 6 33
9.  3_{i}  4_{+} 5_{k} -1 2 122 1^{3} 2_{+} 1  1^{6} 5_{k} 19∗  6 222 33 33 6  6
C_{6}  9  474 888 881 1^{3} 2_{+} 1  6 6 6 33  6 222

Discriminant class: −7

∞  7 19 a_{1} a_{2} a_{3}  Over p = 7  Over p = 19 2 3 5 11 13 17 23 29
G  h  a_{4} a_{5} a_{6} f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
10.  1_{k}  1_{i} 4_{+} 1  -3  -5 2 0_{+} ∗ ∗ 2^{3} 4_{+} ∗  33 6 6 22 6 6 33 33
A_{4} × C_{2}  -3  1  1 1^{2} 1_{i} 7∗ 7∗  2 22 6 33 6 33 6 6
11.  3_{i}  5_{k} 2_{+} -1  8  -1 1^{6} 5_{k} 7∗  2^{2} 2_{+} 1 V 22 222 6 33 222 222 33 22
D_{6}  3  15  6  8  2 0_{+} ∗ ∗ 222 22 222 33 6 22 222 6
12.  3_{i}  3_{k} 4_{+} 1  -7  5 1^{2} 1_{i} 7∗ 7∗ 2^{3} 4_{+} ∗  33 6 6  6 6 33 33
C_{6}  4  53 -97 121 1^{2} 1_{i} 7∗ 7∗  222  6 33 6 33 6 6
13.  3_{i}  5_{k} 4_{+} 1  1 99 1^{6} 5_{k} 7∗  2^{3} 4_{+} ∗  6 222 33 6 6 33 33
C_{6}  39  414 1156 1912  6 33 6 33 6  6 6
14.  3_{i}  5_{k} 4_{+} -1  1 -99 1^{6} 5_{k} 7∗  2^{3} 4_{+} ∗  33 222 6 33 6 222  33
C_{6}  3  281 573 981  6 33 6 33 222 33 222 222

Discriminant class: 1

∞ 7 19 a_{1} a_{2} a_{3} Over p = 7  Over p = 19 2 3 5 11 13 17 23 29
G h  a_{4} a_{5} a_{6} f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
15. 2_{−} 2_{−} 4_{+} -2 0 5 1^{2} 1_{k} 7 7 1^{3} 2_{+} 1  33 33 33 22 33 33 33 33
A_{4} 2  0 5 -1 1^{2} 1_{k} 7 7 1^{3} 2_{+} 1  22 22 33 33 33 33 33 33

Discriminant class: 133

∞  7 19 a_{1} a_{2} a_{3}  Over p = 7  Over p = 19 2 3 5 11 13 17 23 29
G  h  a_{4} a_{5} a_{6} f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
16.  2_{−}  1_{i} 5_{i}  1  2 -8 2 0_{+} ∗ ∗ 1^{6} 5_{i} 19  6 33 6 22 33 6 33 6
A_{4} × C_{2}  -20 -14  7 1^{2} 1_{i} 7∗ 7∗  22 2 33 33 6 6 33 6
17.  2_{−}  5_{k} 3_{k}  3 -12 -29 1^{6} 5_{k} 7∗  2^{2} 2_{+} 1 V 222 22 6 33 22 222 33 222
D_{6}  3  4 19 -13  1^{2} 1_{k} 19 19 22 222 22 33 6 222 22 6
18.  0_{+}  5_{k} 3_{i}  1 -34 -34 1^{6} 5_{k} 7∗  3^{2} 3_{i} 19  6 33 6 33  6 33 222
C_{6}  316 316 -559  33 6  6 6 33 6
19.  0_{+}  3_{k} 5_{i} -1 -36 -30 1^{2} 1_{i} 7∗ 7∗ 1^{6} 5_{i} 19  6 33 6  33 6 33 6
C_{6}  151 147  7 1^{2} 1_{i} 7∗ 7∗  222 33 33 6 6 33 6
20.  0_{+}  5_{k} 5_{i}  1 -55 -160 1^{6} 5_{k} 7∗  1^{6} 5_{i} 19  222 33 222 33 33 6 33 6
C_{6}  3  -20 176 64  33 6 33 33 6 222 33 6
21.  0_{+}  5_{k} 5_{i}  1 -55 -160 1^{6} 5_{k} 7∗  1^{6} 5_{i} 19  6  6 33 33 222  6
C_{6}  3  246 1107 729  33 6 33 33 222 6  222