Low Degree Fields Ramifying Above {2, 3, 13}

Discriminant class: −78

∞ 2 3 13 a_{1} a_{2} a_{3} Over p = 2  Over p = 3  Over p = 13  5 7 11 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_{+} 1_{k} 1_{−} 0 78  1^{2} 3_{+} 2 2 1^{2} 1_{k} 3 3 1^{2} 1_{−} 13∗ 13∗ 2 2 2 2  2  2
C_{2} 4  2  2 2
2. 1_{k} 3_{+} 3_{k} 1_{−} 0 3 -10 1^{2} 3_{+} 2 2 1^{3} 3_{k} 3  1^{2} 1_{−} 13∗ 13∗ 2 2 2 2 3 2 3 2
S_{3} 3 3 2 3 2 2

Discriminant class: −13

∞ 2 3 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 13 5 7 11 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} 4_{+} 1_{+} 0 9 -36 1^{2} 2_{i} −∗ −∗ 1^{3} 4_{+} ∗  1^{2} 1_{+} 13 13 2 3 3  3 2 3
S_{3}  2 2 2  3 3 3

Discriminant class: −3

∞ 2 3 13 a_{1} a_{2} a_{3} Over p = 2 Over p = 3 Over p = 13 5 7 11 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
4. 1_{k} 2_{+} 1_{i} 2_{+} 0 0 -26 1^{3} 2_{+} ∗ 1^{2} 1_{i} 3∗ 3∗ 1^{3} 2_{+} 1 2 3 2 2 2 2 3
S_{3} 3  2 3 2 2 2 3 3
5. 1_{k} 2_{+} 3_{i} 2_{+} 0 0 -52 1^{3} 2_{+} ∗ 1^{3} 3_{i} 3∗ 1^{3} 2_{+} 1 2 3 2 2 3 2 2 3
S_{3} 3  3 2 3 2 2 2
6. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -156 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2 3 2 2 3 2 2
S_{3} 3  2  2 2 2
7. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -78 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2  2 2 3 2 2
S_{3} 3  3 2  2 2 2 3 3
8. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -234 1^{3} 2_{+} ∗  1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  2 3 2 2 3 2 2 3
S_{3} 6  3 2 3 2 2 2 3 3
9. 1_{k} 2_{+} 5_{i} 2_{+} 0 0 -468 1^{3} 2_{+} ∗ 1^{3} 5_{i} 3∗ 1^{3} 2_{+} 1 2 2 2 2 2 3
S_{3} 9  3 2 3 2 2 2

Discriminant class: −1

∞ 2 3 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 13 5 7 11 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
10. 1_{k} 2_{i} 4_{+} 2_{+} 0 -39 -130 1^{2} 2_{i} −1 −1 1^{3} 4_{+} ∗  1^{3} 2_{+} 1  2 2 3 2 2 3 2
S_{3} 3  3 2 2 3 2 3 2
11. 1_{k} 2_{i} 4_{+} 2_{+} 0 -39 -104 1^{2} 2_{i} −1 −1 1^{3} 4_{+} ∗  1^{3} 2_{+} 1  3 2 2  2 2  2
S_{3} 6  3 3 2 2 3 2  2

Discriminant class: 3

∞ 2 3 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 13 5 7 11 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. 0_{+} 2_{i} 3_{k} 2_{+} 0 -39 -78 1^{2} 2_{i} −∗ −∗ 1^{3} 3_{k} 3  1^{3} 2_{+} 1  2 2 3 2 2 3 2 2
S_{3} 3  2 2 3 2 3  2

Discriminant class: 13

∞ 2 3 13 a_{1} a_{2} a_{3} Over p = 2  Over p = 3  Over p = 13 5 7 11 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
13. 0_{+} 2_{+} 4_{+} 1_{+} 0 -12 -10 1^{3} 2_{+} ∗ 1^{3} 4_{+} 1 1^{2} 1_{+} 13 13 2 2 2 3 2 3 2
S_{3} 3  2 2 3 2 3 2 3 2

Discriminant class: 39

∞ 2 3 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 13 5 7 11 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
14. 0_{+} 2_{i} 1_{k} 1_{+} 0 -39  1^{2} 2_{i} −1 −1 1^{2} 1_{k} 3 3 1^{2} 1_{+} 13 13  2 2  2
C_{2} 2  2  2 2 2 2

Discriminant class: 78

∞ 2 3 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 3  Over p = 13  5 7 11 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
15. 0_{+} 3_{i} 1_{i} 1_{−} 0 -78  1^{2} 3_{i} −2 −2 1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{−} 13∗ 13∗ 2  2 2
C_{2} 2  2  2 2
16. 0_{+} 3_{i} 3_{i} 1_{−} 0 -9 -2 1^{2} 3_{i} −2 −2 1^{3} 3_{i} 3∗  1^{2} 1_{−} 13∗ 13∗ 2 3 3 2 2 3 3 3
S_{3}  3  3 2 3 3 2 2