Low Degree Fields Ramifying Above {2, 3, 7}

Discriminant class: −42

∞ 2 3 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 7 5 11 13 17 19 23 29 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_{k} 1_{k} 1_{k} 0 42  1^{2} 3_{k} −2∗ −2∗ 1^{2} 1_{k} 3 3 1^{2} 1_{k} 7 7 2 2  2
C_{2} 4  2  2
2. 1_{k} 3_{k} 3_{k} 1_{k} 0 -6 -16 1^{2} 3_{k} −2∗ −2∗ 1^{3} 3_{k} 3  1^{2} 1_{k} 7 7 2 2 3 3 2 3 3 3
S_{3} 2 3 2 3 3 3

Discriminant class: −21

∞ 2 3 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 7 5 11 13 17 19 23 29 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
3. 1_{k} 2_{i} 1_{i} 1_{k} 0 21  1^{2} 2_{i} −∗ −∗ 1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{k} 7 7  2  2
C_{2} 4  2 2 2 2 2 2
4. 1_{k} 2_{i} 3_{i} 1_{k} 0 -6 -12 1^{2} 2_{i} −∗ −∗ 1^{3} 3_{i} 3∗ 1^{2} 1_{k} 7 7 3 3 2 3 3 2 3
S_{3}  3 3 2 2 2 2 2 2
5. 1_{k} 2_{i} 5_{i} 1_{k} 0 9 -12 1^{2} 2_{i} −∗ −∗ 1^{3} 5_{i} 3∗ 1^{2} 1_{k} 7 7 3 2 3 3 2 3
S_{3}  3 3 2 2 2 2 2 2
6. 1_{k} 2_{i} 5_{i} 1_{k} 0 9 -30 1^{2} 2_{i} −∗ −∗ 1^{3} 5_{i} 3∗ 1^{2} 1_{k} 7 7 3 2 3 3 3 2 3
S_{3}  3 3 2 2 2 2 2 2
7. 1_{k} 2_{i} 5_{i} 1_{k} 0 18 -12 1^{2} 2_{i} −∗ −∗ 1^{3} 5_{i} 3∗ 1^{2} 1_{k} 7 7 3 3 2 3 3 2
S_{3}  2 2 2 2 2 2

Discriminant class: −14

∞ 2 3 7 a_{1} a_{2} a_{3} Over p = 2 Over p = 3 Over p = 7 5 11 13 17 19 23 29 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
8. 1_{k} 3_{+} 4_{+} 1_{i} 0 -3 -26 1^{2} 3_{+} 2 2 1^{3} 4_{+} 1  1^{2} 1_{i} 7∗ 7∗ 3 2  2 3 3 2 2
S_{3} 3  2 2 2 2 2  3 2

Discriminant class: −6

∞ 2 3 7 a_{1} a_{2} a_{3} Over p = 2 Over p = 3 Over p = 7 5 11 13 17 19 23 29 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
9. 1_{k} 3_{−} 1_{k} 2_{+} -1 -2 -6 1^{2} 3_{−} 2∗ 2∗ 1^{2} 1_{k} 3 3 1^{3} 2_{+} 1  3 3 2 2 2 2 3 3
S_{3} 3  2 2 2 2 3 3 2 2
10. 1_{k} 3_{−} 3_{k} 2_{+} 0 -21 -42 1^{2} 3_{−} 2∗ 2∗ 1^{3} 3_{k} 3 1^{3} 2_{+} 1 3 2 2 2 2 3
S_{3} 3  2 2 2 2 3 3 2 2
11. 1_{k} 3_{−} 3_{k} 2_{+} 0 -21 -70 1^{2} 3_{−} 2∗ 2∗ 1^{3} 3_{k} 3  1^{3} 2_{+} 1  3 2 2 2 2 3 3
S_{3} 3  2 2 2 2  3 2 2

Discriminant class: −3

∞ 2 3 7 a_{1} a_{2} a_{3} Over p = 2  Over p = 3  Over p = 7 5 11 13 17 19 23 29 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
12. 1_{k} 2_{+} 1_{i} 2_{+} -1 5  1 1^{3} 2_{+} ∗  1^{2} 1_{i} 3∗ 3∗ 1^{3} 2_{+} 1  2 2 3 2 3 2 2 3
S_{3} 3  3 2 3 2 2 2  3
13. 1_{k} 2_{+} 3_{i} 2_{+} 0 0 -14 1^{3} 2_{+} ∗  1^{3} 3_{i} 3∗  1^{3} 2_{+} 1  2 2  2 3 2 2 3
S_{3} 3  2 3 2 2 2 3
14. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -126 1^{3} 2_{+} ∗ 1^{3} 5_{i} 3∗ 1^{3} 2_{+} 1 2 2 3 2 2 2
S_{3} 9  3 2 3 2 2 2 3
15. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -42 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2 2 3 2 3 2 2 3
S_{3} 3  3 2  2 2 2 3
16. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -84 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2 2 3 2  2 2 3
S_{3} 3  2  2 2 2  3
17. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -252 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2 2  2 3 2 2
S_{3} 6  3 2 3 2 2 2  3

Discriminant class: 2

∞ 2 3 7 a_{1} a_{2} a_{3} Over p = 2  Over p = 3  Over p = 7 5 11 13 17 19 23 29 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
18. 0_{+} 3_{+} 4_{+} 2_{+} 0 -21 -14 1^{2} 3_{+} 2 2 1^{3} 4_{+} ∗ 1^{3} 2_{+} 1 2 2 2 2 3 2 3
S_{3} 3  2 3 2 3 2 2 2 2

Discriminant class: 21

∞ 2 3 7 a_{1} a_{2} a_{3} Over p = 2  Over p = 3  Over p = 7  5 11 13 17 19 23 29 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_{+} 2_{+} 3_{k} 1_{i} 0 -6 -2 1^{3} 2_{+} ∗ 1^{3} 3_{k} 3 1^{2} 1_{i} 7∗ 7∗ 3 2 2 3 2 2 2 2
S_{3}  3 3 3 3 2 3 2

Discriminant class: 42

∞ 2 3 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 7  5 11 13 17 19 23 29 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
20. 0_{+} 3_{−} 1_{i} 1_{i} 0 -42  1^{2} 3_{−} 2∗ 2∗ 1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{i} 7∗ 7∗ 2  2  2
C_{2} 2  2  2  2  2
21. 0_{+} 3_{−} 5_{i} 1_{i} 0 -27 -30 1^{2} 3_{−} 2∗ 2∗ 1^{3} 5_{i} 3∗  1^{2} 1_{i} 7∗ 7∗ 2 3 3 3  2 3 2
S_{3}  2 3 2 3 3 2 3 2