Low Degree Fields Ramifying Above {3, 53}

Discriminant class: −159

∞ 3 53 a_{1} a_{2} a_{3} Over p = 3  Over p = 53  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 43 47 59 61
1. 1_{k} 1_{k} 1_{−} 0 159  1^{2} 1_{k} 3 3 1^{2} 1_{−} 53∗ 53∗  2  2 2
C_{2} 10  2 2  2 2 2
2. 1_{k} 3_{k} 1_{−} 0 3 -7 1^{3} 3_{k} 3  1^{2} 1_{−} 53∗ 53∗ 3 3  2 3 2 2
S_{3}  2 2 3 3  2 2 2

Discriminant class: −3

∞ 3 53 a_{1} a_{2} a_{3} Over p = 3 Over p = 53 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 43 47 59 61
3. 1_{k} 1_{i} 2_{+} 0 0 -53 1^{2} 1_{i} 3∗ 3∗ 1^{3} 2_{+} ∗ 2 2 3 2 2 3 2
S_{3}  2 3 3 2 3 2 2
4. 1_{k} 5_{i} 2_{+} 0 0 -477 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  2 2  2 3 2 3 2
S_{3} 9  2 3 3 2  2 2
5. 1_{k} 5_{i} 2_{+} 0 0 -159 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  2 2 3 2 3 2  2
S_{3} 3  2  2 3 2 2