Low Degree Fields Ramifying Above {2, 3, 29}

Discriminant class: −174

∞ 2 3 29 a_{1} a_{2} a_{3} Over p = 2 Over p = 3 Over p = 29 5 7 11 13 17 19 23 31
G h  f^{e} c_{w} d s f^{e} c_{w} d s f^{e} c_{w} d s 37 41 43 47 53 59 61 67
1. 1_{k} 3_{+} 1_{i} 1_{+} 0 174 1^{2} 3_{+} 2 2 1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{+} 29 29 2 2 2 2
C_{2} 12  2
2. 1_{k} 3_{+} 1_{i} 1_{+} -1 -2  6 1^{2} 3_{+} 2 2 1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{+} 29 29 3 3 2 2 3 3 2 2
S_{3}  3 3 3 3  3  2
3. 1_{k} 3_{+} 5_{i} 1_{+} 0 -18 -96 1^{2} 3_{+} 2 2 1^{3} 5_{i} 3∗  1^{2} 1_{+} 29 29 3 3 2 2  3 2 2
S_{3} 3  3  3 3 3  2
4. 1_{k} 3_{+} 5_{i} 1_{+} 0 27 -126 1^{2} 3_{+} 2 2 1^{3} 5_{i} 3∗  1^{2} 1_{+} 29 29 3  2 2 3 3 2 2
S_{3} 3  3 3  3 3  2
5. 1_{k} 3_{+} 5_{i} 1_{+} 0 27 -222 1^{2} 3_{+} 2 2 1^{3} 5_{i} 3∗  1^{2} 1_{+} 29 29  3 2 2 3  2 2
S_{3} 12  3 3 3 3 3  2

Discriminant class: −58

∞ 2 3 29 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 29  5 7 11 13 17 19 23 31
G h  f^{e} c_{w} d  s f^{e} c_{w} d s f^{e} c_{w} d  s 37 41 43 47 53 59 61 67
6. 1_{k} 3_{k} 4_{+} 1_{−} 0 15 -14 1^{2} 3_{k} −2∗ −2∗ 1^{3} 4_{+} ∗  1^{2} 1_{−} 29∗ 29∗ 2 2 2 2 2 2 2
S_{3}  3 2 2 3 2 3 3 3

Discriminant class: −3

∞ 2 3 29 a_{1} a_{2} a_{3} Over p = 2 Over p = 3 Over p = 29 5 7 11 13 17 19 23 31
G h  f^{e} c_{w} d s f^{e} c_{w} d s f^{e} c_{w} d s 37 41 43 47 53 59 61 67
7. 1_{k} 2_{+} 1_{i} 2_{+} -1 -19 73 1^{3} 2_{+} ∗ 1^{2} 1_{i} 3∗ 3∗ 1^{3} 2_{+} ∗ 2 3 2 2 3 2
S_{3}  3 2 3 2 2 2 3 3
8. 1_{k} 2_{+} 3_{i} 2_{+} 0 0 -58 1^{3} 2_{+} ∗ 1^{3} 3_{i} 3∗ 1^{3} 2_{+} ∗ 2 3 2 3 2 2
S_{3} 6  3 2 3 2 2 2
9. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -348 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  2 3 2 3 2 3 2 3
S_{3} 3  3 2  2 2 2 3 3
10. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -522 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  2 3 2 3 2 3 2 3
S_{3} 3  3 2 3 2 2 2
11. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -174 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  2  2  2 3 2 3
S_{3} 12  2  2 2 2
12. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -1044 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  2  2 3 2  2 3
S_{3} 3  2 3 2 2 2 3 3

Discriminant class: −2

∞ 2 3 29 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 29 5 7 11 13 17 19 23 31
G h  f^{e} c_{w} d  s f^{e} c_{w} d s f^{e} c_{w} d s 37 41 43 47 53 59 61 67
13. 1_{k} 3_{i} 4_{+} 2_{+} 0 87 -638 1^{2} 3_{i} −2 −2 1^{3} 4_{+} 1  1^{3} 2_{+} ∗  2 2  2 3  2 2
S_{3} 3  2  2 2 3 2 3

Discriminant class: −2 (continued)

∞ 2 3 29 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 29 5 7 11 13 17 19 23 31
G h  f^{e} c_{w} d  s f^{e} c_{w} d s f^{e} c_{w} d s 37 41 43 47 53 59 61 67
14. 1_{k} 3_{i} 4_{+} 2_{+} 0 -174 -928 1^{2} 3_{i} −2 −2 1^{3} 4_{+} 1  1^{3} 2_{+} ∗  2 2 3 2 3 3 2 2
S_{3} 6  2 3 3 2 2 3 2 3

Discriminant class: 3

∞ 2 3 29 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 29 5 7 11 13 17 19 23 31
G h  f^{e} c_{w} d  s f^{e} c_{w} d s f^{e} c_{w} d s 37 41 43 47 53 59 61 67
15. 0_{+} 2_{i} 3_{k} 2_{+} 0 -87 116 1^{2} 2_{i} −∗ −∗ 1^{3} 3_{k} 3  1^{3} 2_{+} ∗  2 2 3 3 2 2 3 2
S_{3}  3 2 2  2  3 2

Discriminant class: 58

∞ 2 3 29 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 29  5 7 11 13 17 19 23 31
G h  f^{e} c_{w} d s f^{e} c_{w} d s f^{e} c_{w} d  s 37 41 43 47 53 59 61 67
16. 0_{+} 3_{−} 4_{+} 1_{−} 0 -21 -26 1^{2} 3_{−} 2∗ 2∗ 1^{3} 4_{+} 1  1^{2} 1_{−} 29∗ 29∗ 2 3 3 2 2 3  2
S_{3} 3  2 3 2 2 2 3 2

Discriminant class: 87

∞ 2 3 29 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 29  5 7 11 13 17 19 23 31
G h  f^{e} c_{w} d  s f^{e} c_{w} d s f^{e} c_{w} d  s 37 41 43 47 53 59 61 67
17. 0_{+} 2_{i} 1_{i} 1_{−} 0 -87  1^{2} 2_{i} −1 −1 1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{−} 29∗ 29∗ 2 2 2
C_{2} 2 2 2 2 2 2
18. 0_{+} 2_{i} 3_{i} 1_{−} 0 -15 -6 1^{2} 2_{i} −1 −1 1^{3} 3_{i} 3∗  1^{2} 1_{−} 29∗ 29∗ 2 2 2 3 3 3 3 3
S_{3}  2 3 3 2 2 3 2 2

Discriminant class: 174

∞ 2 3 29 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 29 5 7 11 13 17 19 23 31
G h  f^{e} c_{w} d  s f^{e} c_{w} d s f^{e} c_{w} d s 37 41 43 47 53 59 61 67
19. 0_{+} 3_{i} 1_{k} 1_{+} 0 -174  1^{2} 3_{i} −2 −2 1^{2} 1_{k} 3 3 1^{2} 1_{+} 29 29  2  2  2
C_{2} 2  2 2  2