Old Sextic Fields Unramified Outside of {∞, 5, 19}

Discriminant class: −95

∞ 5 19 a_{1} a_{2} a_{3} Over p = 5  Over p = 19 2 3 7 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} 3_{+} 5_{k} 1 21 -27 3^{2} 3_{+} 5  1^{6} 5_{k} 19∗  33 33 222  33 6 6 6
C_{6} 8  37 -33 121  222  6 6 6 33 6 33

Discriminant class: −19

∞ 5 19 a_{1}  a_{2} a_{3} Over p = 5  Over p = 19 2 3 7 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
2. 3_{i} 0_{+} 5_{k} -1  2  8 3 0_{+} 1  1^{6} 5_{k} 19∗  6 6  6 33 33 6
C_{6}  -1  -5  7 3 0_{+} 1  222 222 6 33 33 6 6 33
3. 3_{i} 8_{+} 5_{k}  3 -20 -45 1^{5} 8_{+} 1  1^{6} 5_{k} 19∗  222 4 5 33 4 5 5 4
S_{5} 48  660 -2661 6547  222 4 6 33 22 4 4 5
4. 3_{i} 8_{+} 5_{k}  3 -20 -45 1^{5} 8_{+} 1  1^{6} 5_{k} 19∗  6 4 5 22 222 5 33 4
S_{5} 24  660 -1293 1531  4 6 4 33 33 6 6 22
5. 3_{i} 8_{+} 5_{k} -3 -20 -50 1^{5} 8_{+} 1  1^{6} 5_{k} 19∗  6 6 22 33 4 22 5 6
S_{5} 24  375 1635 1645  4 4 222 33 22 4 4 5

Discriminant class: 5

∞ 5 19 a_{1}  a_{2} a_{3} Over p = 5  Over p = 19 2 3 7 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
6. 0_{+} 3_{+} 4_{+} -1 -16  1 3^{2} 3_{+} 5  1^{3} 2_{+} 1  6 6 222  6 6 6 33
C_{6}  47  10 -11  1^{3} 2_{+} 1  222 33 6 6 6 33 33
7. 2_{−} 5_{−} 4_{+} -2  0 10 1^{5} 5_{−} 5  1^{5} 4_{+} 1  222 6 6 5 4 4 222 5
S_{5}  -10  4 17  33 4 33 4 4 6 5 33
8. 2_{−} 7_{−} 4_{+} -2  -5 20 1^{5} 7_{−} 5  1^{5} 4_{+} 1  6 222 4 33 6 6 4 5
S_{5}  -25  -5 35  33 222 22 6 6 222 22 5
9. 2_{−} 7_{−} 4_{+} -1  0 -35 1^{5} 7_{−} 5  1^{3} 2_{+} 1  4 222 6 33 6 4 4 5
S_{5} 3  0 -61 46  1^{3} 2_{+} 1  5 4 5 6 6 4 5 5
10. 2_{−} 9_{−} 4_{+} -1 -25 -50 1^{5} 9_{−} 5  1^{3} 2_{+} 1  222 6 6 5 6 6 222 33
S_{5} 3  475 -1805 1805  1^{3} 2_{+} 1  5 4 33 4 6 4 22 22
11. 2_{−} 9_{−} 4_{+} 0  25 -150 1^{5} 9_{−} 5  1^{5} 4_{+} 1  6 4 4 33 222 6 6 5
S_{5}  350 -610 375  22 6 22 6 4 6 33 5