Low Degree Fields Ramifying Above {3, 11, 13}

Discriminant class: −143

∞ 3 11 13 a_{1} a_{2} a_{3} Over p = 3  Over p = 11  Over p = 13  2 5 7 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} 4_{+} 1_{k} 1_{−} 0 -9 -63 1^{3} 4_{+} 1  1^{2} 1_{k} 11 11 1^{2} 1_{−} 13∗ 13∗ 3 2  2 3 3 2 2
S_{3} 3  2 3 2 2  2 2 2

Discriminant class: −3

∞ 3 11 13 a_{1} a_{2} a_{3} Over p = 3 Over p = 11 Over p = 13 2 5 7 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
2. 1_{k} 1_{i} 2_{+} 2_{+} 0 0 -143 1^{2} 1_{i} 3∗ 3∗ 1^{3} 2_{+} ∗ 1^{3} 2_{+} 1 2 2 3 2 3 2 2 3
S_{3} 3  3 2 3 2 2 2 3
3. 1_{k} 3_{i} 2_{+} 2_{+} 0 0 -1573 1^{3} 3_{i} 3∗ 1^{3} 2_{+} ∗ 1^{3} 2_{+} 1 2 2 3 2 3 2 2
S_{3} 6  3 2 3 2 2 2 3 3
4. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -1287 1^{3} 5_{i} 3∗ 1^{3} 2_{+} ∗ 1^{3} 2_{+} 1 2 2 2 3 2 2
S_{3} 12  2 3 2 2 2 3
5. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -429 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2 2 3 2  2 2 3
S_{3} 3  3 2  2 2 2 3
6. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -4719 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2 2  2  2 2 3
S_{3} 3  3 2  2 2 2 3 3
7. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -5577 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2 2 3 2 3 2 2 3
S_{3} 9  2 3 2 2 2 3 3

Discriminant class: 429

∞ 3 11 13 a_{1} a_{2} a_{3}  Over p = 3  Over p = 11  Over p = 13  2 5 7 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. 0_{+} 1_{i} 1_{i} 1_{−} 0 -429  1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{i} 11∗ 11∗ 1^{2} 1_{−} 13∗ 13∗ 2  2  2
C_{2} 2  2 2 2  2  2 2
9. 0_{+} 5_{i} 1_{i} 1_{−} 0 -36 -75 1^{3} 5_{i} 3∗  1^{2} 1_{i} 11∗ 11∗ 1^{2} 1_{−} 13∗ 13∗ 2  3 3 3 2 3 2
S_{3}  2 2 2 3 2 3 2 2