Old Sextic Fields Unramified Outside of {∞, 13, 19}

Discriminant class: −247

∞  13 19 a_{1}  a_{2}  a_{3}  Over p = 13  Over p = 19 2 3 5 7 11 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}  3_{−} 3_{i}  2  -6  -1 1^{2} 1_{−} 13∗ 13∗ 1^{2} 1_{k} 19 19 33 222 222 222 222 33 33 222
S_{3}  2  80 -21  9 1^{2} 1_{−} 13∗ 13∗ 1^{2} 1_{k} 19 19 33 33 33 33 222 222 33 33
2.  3_{i}  5_{−} 3_{i}  -1  60  -61 1^{6} 5_{−} 13∗  3^{2} 3_{i} 19  33 6 222 6 6 33 33 6
C_{6}  18  526  62 1949  33 33 33 222 222 33 33
3.  3_{i}  3_{−} 5_{i}  1  59  -65 3^{2} 3_{−} 13∗  1^{6} 5_{i} 19  33 6 6 222 222 33 33 6
C_{6}  18  341 -337 1147  33 33 6 6 33 33
4.  3_{i}  5_{−} 5_{i}  -1  21  -87 1^{6} 5_{−} 13∗  1^{6} 5_{i} 19  222 6 6 6 33 33 6
C_{6}  378  2450 -2304 15872  33  33 6 6 33
5.  3_{i}  5_{−} 5_{i}  -1  21 407 1^{6} 5_{−} 13∗  1^{6} 5_{i} 19  33 6 6 6 6  222
C_{6}  162  -761 -5515 12167  33 33  6 6  33
6.  3_{i}  5_{−} 3_{i}  -2 -20  58 1^{6} 5_{−} 13∗  3^{2} 3_{i} 19  3 6 222 6 6 3 3 6
S_{3} × C_{3} 18  135 -512 404  33 3 3 3 222 222 3 3
7.  3_{i}  3_{−} 5_{i}  1  21  68 3^{2} 3_{−} 13∗  1^{6} 5_{i} 19  3 6 6 222 222 3 3 6
S_{3} × C_{3} 18  341 594 368  33 33 3 3 6 6 3 3
8.  3_{i}  5_{−} 5_{i}  -1  21 160 1^{6} 5_{−} 13∗  1^{6} 5_{i} 19  3 6 6 6 6 33 33 222
S_{3} × C_{3} 162  1709 6588 14884  33 3 3 33 6 6 33 3
9.  3_{i}  5_{−} 5_{i}  -1  21  -87 1^{6} 5_{−} 13∗  1^{6} 5_{i} 19  33 222 6 6 6 3 3 6
S_{3} × C_{3} 54  227 -81 311  33 3 33 3 6 6 3 33
10.  3_{i}  3_{+} 3_{k}  -1  7  5 1^{4} 3_{+} 13∗ 13∗ 2^{2} 2_{+} 1 V 33 4 4 4 222 33 33 222
S_{4}^{−}  2  17  25  9  1^{2} 1_{k} 19 19 33 33 33 33 222 4 33 33

Discriminant class: −19

∞  13 19 a_{1}  a_{2}  a_{3}  Over p = 13  Over p = 19  2 3 5 7 11 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
11.  3_{i}  0_{+} 5_{k}  -1  2  8 6 0_{+} ∗  1^{6} 5_{k} 19∗  6 6 33  33 33 6
C_{6}  -1  -5  7  222 222 6 33 33 6 6 33
12.  3_{i}  4_{+} 3_{k}  1  6  -1 2^{3} 4_{+} ∗  3^{2} 3_{k} 19∗  6 6  33 33 33 33 6
C_{6}  13  85 118 415  222 6 6 33  222 6 33
13.  3_{i}  4_{+} 5_{k}  -1  2  8 2^{3} 4_{+} ∗  1^{6} 5_{k} 19∗  6 6 33 33 33  222
C_{6}  21  1044 -6712 14789  222 6 6  33 6 222 33
14.  3_{i}  4_{+} 5_{k}  -1  2 255 2^{3} 4_{+} ∗  1^{6} 5_{k} 19∗  222 222 33 33 33 33 33 6
C_{6}  3  1291 -784 2192  222 6 222 33 33 6 6
15.  3_{i}  2_{−} 3_{i}  -2  4  -4 2^{2} 2_{−} 1  V 2^{2} 2_{+} 1  V 6 222 22 22 22 33 33 222
D_{6}  8 -11  5 2 0_{+} ∗  ∗ 1^{2} 1_{i} 19∗ 19∗ 6 6 6 33 22 222 6 33
16.  3_{i}  4_{+} 3_{k}  -1  6  10 1^{4} 3_{+} 13∗ 13∗ 1^{2} 1_{k} 19 19 6 4 42 42 22 33 33 222
S_{4} × C_{2} 4  52  10  81 1^{2} 1_{+} 13 13 1^{2} 1_{k} 19 19 6 6 6 33 22 4 6 33
17.  3_{i}  4_{+} 3_{k}  -2  11  -10 1^{5} 4_{+} ∗  1^{4} 3_{k} 19 19∗ 4 4 33 33 33 22 22 6
S_{5}  43 -31  6  2 0_{+} ∗  ∗ 6 4 4 5 5 4 222 5

Discriminant class: 1

∞ 13 19 a_{1} a_{2} a_{3}  Over p = 13  Over p = 19 2 3 5 7 11 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
18. 2_{−} 4_{−} 2_{+} 3 1 -7 1^{4} 3_{+} 13∗ 13∗ 2^{2} 2_{+} 1 V 33 42 42 42 22 33 33 22
S_{4}^{+} 2  -1 3 -1 1^{2} 1_{−} 13∗ 13∗  33 33 33 33 22 42 33 33

Discriminant class: 13

∞  13 19 a_{1} a_{2} a_{3}  Over p = 13  Over p = 19 2 3 5 7 11 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
19.  0_{+}  5_{+} 0_{+}  1  -5  -4 1^{6} 5_{+} 13  6 0_{+} ∗  6 33 222 6 6 33 33 33
C_{6}  6  3  -1  222 6 6 33 222  6 33
20.  0_{+}  3_{+} 4_{+}  -1 -22  5 3^{2} 3_{+} 13  2^{3} 4_{+} ∗  6 33 6 222 222 33 33 33
C_{6}  73 -58  1  222 222 6 33 6 33 6 33
21.  0_{+}  5_{+} 4_{+}  -1 -83 -100 1^{6} 5_{+} 13  2^{3} 4_{+} ∗  222  6 6 6 33 33 33
C_{6}  3  1345 2909 -183  222 6 222 33 6 33 6
22.  0_{+}  5_{+} 4_{+}  -1 -83 -100 1^{6} 5_{+} 13  2^{3} 4_{+} ∗  6 33 6 6 6
C_{6}  3  1592 4391 2287  222 6 6  6 33 222 33
23.  2_{−}  3_{−} 2_{+}  -3  6  -7 2^{2} 2_{−} 1  V 2^{2} 2_{+} 1 V 6 22 222 222 222 33 33 22
D_{6}  7  -4  -1 1^{2} 1_{+} 13 13 2 0_{+} ∗ ∗ 6 6 6 33 222 22 6 33
24.  2_{−}  3_{+} 2_{−}  -3  0  5 1^{4} 3_{+} 13∗ 13∗ 2 0_{+} ∗ ∗ 6 42 4 4 222 33 33 22
S_{4} × C_{2}  1  -4  1 2 0_{+} ∗  ∗ 1^{2} 1_{k} 19 19 6 6 6 33 222 42 6 33