Low Degree Fields Ramifying Above {7, 59}

Discriminant class: −59

∞ 7 59 a_{1} a_{2} a_{3} Over p = 7  Over p = 59  2 3 5 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 53 61
1. 1_{k} 2_{+} 1_{i} -1 5 8 1^{3} 2_{+} 1  1^{2} 1_{i} 59∗ 59∗ 2 3 3 2 2 3  2
S_{3} 3  3 2 2  2 2  2
2. 1_{k} 2_{+} 1_{i} -1 -9 36 1^{3} 2_{+} 1  1^{2} 1_{i} 59∗ 59∗ 2  3 2 2 3 3 2
S_{3} 3  3 2 2 3 2 2 3 2
3. 1_{k} 2_{+} 1_{i} -1 -2 -20 1^{3} 2_{+} 1 1^{2} 1_{i} 59∗ 59∗ 2 3 2 2 3 3 2
S_{3} 3  2 2 3 2 2 3 2

Discriminant class: −7

∞ 7 59 a_{1} a_{2} a_{3} Over p = 7 Over p = 59 2 3 5 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 53 61
4. 1_{k} 1_{i} 2_{+} -1 20 -153 1^{2} 1_{i} 7∗ 7∗ 1^{3} 2_{+} ∗  3 2 2 3 2 2 2 3
S_{3}  3 2 3 2 3 2 3 2

Discriminant class: 413

∞ 7 59 a_{1} a_{2} a_{3} Over p = 7 Over p = 59 2 3 5 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 53 61
5. 0_{+} 1_{i} 1_{k} 0 -413 1^{2} 1_{i} 7∗ 7∗ 1^{2} 1_{k} 59 59 2 2 2 2 2 2 2
C_{2}  2 2 2