Low Degree Fields Ramifying Above {3, 31}

Discriminant class: −3

∞ 3 31 a_{1} a_{2} a_{3} Over p = 3 Over p = 31 2 5 7 11 13 17 19 23
G h  f^{e} c_{w} d s f^{e} c_{w} d s 29 37 41 43 47 53 59 61
1. 1_{k} 3_{i} 2_{+} 0 0 -31 1^{3} 3_{i} 3∗  1^{3} 2_{+} 1  2 2 3 2  2  2
S_{3} 3  2  2 3 2 2 2 3
2. 1_{k} 5_{i} 2_{+} 0 0 -93 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2 2 3 2 3 2 3 2
S_{3} 3  2 3 2 3 2 2 2 3
3. 1_{k} 5_{i} 2_{+} 0 0 -279 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2 2  2 3 2 3 2
S_{3} 3  2 3 2  2 2 2 3

Discriminant class: 1

∞ 3 31 a_{1} a_{2} a_{3} Over p = 3  Over p = 31 2 5 7 11 13 17 19 23
G h  f^{e} c_{w} d s f^{e} c_{w} d s 29 37 41 43 47 53 59 61
4. 0_{+} 4_{+} 2_{+} 0 -93 -341 1^{3} 4_{+} 1  1^{3} 2_{+} 1  3 3 3  3 3 3 3
C_{3} 3  3 3 3 3 3 3  3
5. 0_{+} 4_{+} 2_{+} 0 -93 217 1^{3} 4_{+} 1  1^{3} 2_{+} 1  3  3  3 3 3
C_{3} 3  3 3  3 3 3 3

Discriminant class: 93

∞ 3 31 a_{1} a_{2} a_{3} Over p = 3  Over p = 31  2 5 7 11 13 17 19 23
G h  f^{e} c_{w} d s f^{e} c_{w} d  s 29 37 41 43 47 53 59 61
6. 0_{+} 1_{k} 1_{i} 0 -93  1^{2} 1_{k} 3 3 1^{2} 1_{i} 31∗ 31∗ 2 2  2
C_{2}  2 2 2 2  2 2
7. 0_{+} 3_{k} 1_{i} 0 -6 -1 1^{3} 3_{k} 3  1^{2} 1_{i} 31∗ 31∗ 2 2 3 3 2 3 3 3
S_{3}  3 2 2 2 2  2 2