Low Degree Fields Ramifying Above {3, 7, 29}

Discriminant class: −203

∞ 3 7 29 a_{1} a_{2} a_{3} Over p = 3 Over p = 7 Over p = 29 2 5 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} 4_{+} 1_{i} 1_{+} 0 15 -44 1^{3} 4_{+} 1  1^{2} 1_{i} 7∗ 7∗ 1^{2} 1_{+} 29 29 2 2 2 2 3 3 3 3
S_{3} 3 2 2 3 3 2 3

Discriminant class: −87

∞ 3 7 29 a_{1} a_{2} a_{3} Over p = 3  Over p = 7  Over p = 29  2 5 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
2. 1_{k} 3_{k} 2_{+} 1_{−} 0 -21 -119 1^{3} 3_{k} 3  1^{3} 2_{+} 1  1^{2} 1_{−} 29∗ 29∗ 3 2  3  2 2 2
S_{3} 3  2 3 2 3 2 2 2 3
3. 1_{k} 3_{k} 2_{+} 1_{−} 0 21 -7 1^{3} 3_{k} 3  1^{3} 2_{+} 1  1^{2} 1_{−} 29∗ 29∗ 3 2 3 3 3 2 2 2
S_{3} 3  2 3 2  2 2 2
4. 1_{k} 3_{k} 2_{+} 1_{−} 0 -21 -84 1^{3} 3_{k} 3  1^{3} 2_{+} 1  1^{2} 1_{−} 29∗ 29∗  2 3  3 2 2 2
S_{3} 6  2 3 2 3 2 2 2 3

Discriminant class: −3

∞ 3 7 29 a_{1} a_{2} a_{3} Over p = 3 Over p = 7 Over p = 29 2 5 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
5. 1_{k} 1_{i} 2_{+} 2_{+} -1 68 -188 1^{2} 1_{i} 3∗ 3∗ 1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 2 3 2 3 2 3
S_{3} 6  3 2  2 2 2 3
6. 1_{k} 3_{i} 2_{+} 2_{+} 0 0 -203 1^{3} 3_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 2  2 3 2 3
S_{3} 9  3 2 3 2 2 2  3
7. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -609 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 2 3 2  2 3
S_{3} 3  3 2 3 2 2 2  3
8. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -12789 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 2 3 2 3 2 3
S_{3} 3  3 2 3 2 2 2 3
9. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -1827 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 2 3 2 3 2
S_{3} 3  2  2 2 2  3
10. 1_{k} 5_{i} 2_{+} 2_{+} 0 0 -4263 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 2  2  2
S_{3} 6  2 3 2 2 2 3

Discriminant class: 21

∞ 3 7 29 a_{1} a_{2} a_{3} Over p = 3 Over p = 7 Over p = 29 2 5 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
11. 0_{+} 1_{k} 1_{i} 2_{+} -1 -19 -14 1^{2} 1_{k} 3 3 1^{2} 1_{i} 7∗ 7∗ 1^{3} 2_{+} ∗  2 3 2 2 3 2 2 2
S_{3}  3 3 3  2 3 2 3

Discriminant class: 609

∞ 3 7 29 a_{1} a_{2} a_{3}  Over p = 3  Over p = 7  Over p = 29  2 5 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
12. 0_{+} 1_{i} 1_{i} 1_{−} 0 -609  1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{i} 7∗ 7∗ 1^{2} 1_{−} 29∗ 29∗  2 2  2
C_{2} 2  2 2 2 2 2
13. 0_{+} 5_{i} 1_{i} 1_{−} 0 -27 -33 1^{3} 5_{i} 3∗  1^{2} 1_{i} 7∗ 7∗ 1^{2} 1_{−} 29∗ 29∗ 3 3  2 2 3 2 3
S_{3}  2 2 2 2 2  3 3