Low Degree Fields Ramifying Above {3, 23, 29}

Discriminant class: −667

∞ 3 23 29 a_{1} a_{2} a_{3} Over p = 3  Over p = 23  Over p = 29 2 5 7 11 13 17 19 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} 4_{+} 1_{i} 1_{+} 0 -39 -202 1^{3} 4_{+} ∗  1^{2} 1_{i} 23∗ 23∗ 1^{2} 1_{+} 29 29 2 2 2  3  3 2
S_{3}  3 2  2 2 3 3 2

Discriminant class: −87

∞ 3 23 29 a_{1} a_{2} a_{3} Over p = 3  Over p = 23  Over p = 29  2 5 7 11 13 17 19 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
2. 1_{k} 3_{k} 2_{+} 1_{−} 0 69 -299 1^{3} 3_{k} 3  1^{3} 2_{+} ∗  1^{2} 1_{−} 29∗ 29∗ 3 2 3 3  2 2
S_{3} 3  2  2  2 2 2 3
3. 1_{k} 3_{k} 2_{+} 1_{−} 0 69 966 1^{3} 3_{k} 3  1^{3} 2_{+} ∗  1^{2} 1_{−} 29∗ 29∗  2  3 3 3 2 2
S_{3} 3  2  2 3 2 2 2
4. 1_{k} 3_{k} 2_{+} 1_{−} 0 -69 -253 1^{3} 3_{k} 3  1^{3} 2_{+} ∗  1^{2} 1_{−} 29∗ 29∗ 3 2 3  3 3 2 2
S_{3} 3  2  2 3 2 2 2 3

Discriminant class: −3

∞ 3 23 29 a_{1} a_{2} a_{3} Over p = 3 Over p = 23 Over p = 29 2 5 7 11 13 17 19 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
5. 1_{k} 1_{i} 2_{+} 2_{+} 0 0 -667 1^{2} 1_{i} 3∗ 3∗ 1^{3} 2_{+} ∗  1^{3} 2_{+} ∗  2 2 3 2 3 2 3
S_{3} 19  2  2 2 2 3
6. 1_{k} 3_{i} 2_{+} 2_{+} 0 0 -15341 1^{3} 3_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} ∗  2 2 3 2  2
S_{3} 3  2 3 2 2 2 3 3
7. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -46023 1^{3} 5_{i} 3∗ 1^{3} 2_{+} ∗ 1^{3} 2_{+} ∗ 2 2 3 2 3 2 3 3
S_{3} 24  3 2 3 2 2 2 3 3
8. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -2001 1^{3} 5_{i} 3∗ 1^{3} 2_{+} ∗ 1^{3} 2_{+} ∗ 2 2 2 2 3 3
S_{3} 84  3 2 3 2 2 2 3
9. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -6003 1^{3} 5_{i} 3∗ 1^{3} 2_{+} ∗ 1^{3} 2_{+} ∗ 2 2 3 2 3 2 3
S_{3} 3  3 2 3 2 2 2 3
10. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -58029 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} ∗  2 2  2 3 2 3 3
S_{3} 3  3 2  2 2 2 3 3

Discriminant class: 69

∞ 3 23 29 a_{1} a_{2} a_{3} Over p = 3 Over p = 23 Over p = 29 2 5 7 11 13 17 19 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
11. 0_{+} 5_{i} 1_{k} 2_{+} 0 -522 -87 1^{3} 5_{i} 3∗ 1^{2} 1_{k} 23 23 1^{3} 2_{+} ∗ 2 3 2 3 2
S_{3}  2 2 2 2 3 2 2 2
12. 0_{+} 5_{i} 1_{k} 2_{+} 0 -261 -1392 1^{3} 5_{i} 3∗  1^{2} 1_{k} 23 23 1^{3} 2_{+} ∗  2 3 2 3 3 3 2 3
S_{3}  2 2 2 2  2 2 2
13. 0_{+} 5_{i} 1_{k} 2_{+} 0 -522 -4089 1^{3} 5_{i} 3∗  1^{2} 1_{k} 23 23 1^{3} 2_{+} ∗  2  2 3 3  2 3
S_{3}  2 2 2 2 3 2 2 2

Discriminant class: 2001

∞ 3 23 29 a_{1}  a_{2} a_{3} Over p = 3  Over p = 23  Over p = 29  2 5 7 11 13 17 19 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. 0_{+} 1_{k} 1_{k} 1_{−} 0 -2001  1^{2} 1_{k} 3 3 1^{2} 1_{k} 23 23 1^{2} 1_{−} 29∗ 29∗  2 2  2  2
C_{2} 2  2  2