Low Degree Fields Ramifying Above {3, 43}

Discriminant class: −43

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

Discriminant class: −3

∞ 3 43 a_{1} a_{2} a_{3} Over p = 3 Over p = 43 2 5 7 11 13 17 19 23
G h  f^{e} c_{w} d s f^{e} c_{w} d s 29 31 37 41 47 53 59 61
2. 1_{k} 3_{i} 2_{+} 0 0 -43 1^{3} 3_{i} 3∗  1^{3} 2_{+} 1  2 2  2 3 2 3 2
S_{3} 12  2 3  2 2 2 2 3
3. 1_{k} 5_{i} 2_{+} 0 0 -129 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2 2 3 2  2 3 2
S_{3} 6  2 3 3 2 2 2 2 3
4. 1_{k} 5_{i} 2_{+} 0 0 -387 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2 2 3 2 3 2  2
S_{3} 3  2  3 2 2 2 2 3

Discriminant class: 1

∞ 3 43 a_{1} a_{2} a_{3} Over p = 3  Over p = 43 2 5 7 11 13 17 19 23
G h  f^{e} c_{w} d s f^{e} c_{w} d s 29 31 37 41 47 53 59 61
5. 0_{+} 4_{+} 2_{+} 0 -129 -215 1^{3} 4_{+} 1 1^{3} 2_{+} 1 3 3 3 3 3
C_{3} 3  3 3 3 3 3
6. 0_{+} 4_{+} 2_{+} 0 -129 -559 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 3

Discriminant class: 129

∞ 3 43 a_{1} a_{2} a_{3} Over p = 3  Over p = 43  2 5 7 11 13 17 19 23
G h  f^{e} c_{w} d s f^{e} c_{w} d  s 29 31 37 41 47 53 59 61
7. 0_{+} 1_{k} 1_{i} 0 -129  1^{2} 1_{k} 3 3 1^{2} 1_{i} 43∗ 43∗  2 2  2 2 2
C_{2}  2 2 2 2 2 2