Low Degree Fields Ramifying Above {11, 13, 29}

Discriminant class: −4147

∞ 11 13 29 a_{1} a_{2} a_{3}  Over p = 11  Over p = 13  Over p = 29  2 3 5 7 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} 1_{i} 1_{−} 1_{−} 0 4147  1^{2} 1_{i} 11∗ 11∗ 1^{2} 1_{−} 13∗ 13∗ 1^{2} 1_{−} 29∗ 29∗ 2 2 2  2
C_{2} 12  2  2  2
2. 1_{k} 1_{i} 1_{−} 1_{−} -1  -1 -12 1^{2} 1_{i} 11∗ 11∗ 1^{2} 1_{−} 13∗ 13∗ 1^{2} 1_{−} 29∗ 29∗ 2 2 2 3 3 2 3 3
S_{3} 3 2 3 3 2 3 2

Discriminant class: −143

∞ 11 13 29 a_{1} a_{2} a_{3}  Over p = 11  Over p = 13  Over p = 29 2 3 5 7 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
3. 1_{k} 1_{k} 1_{−} 2_{+} 0 29 -29 1^{2} 1_{k} 11 11 1^{2} 1_{−} 13∗ 13∗ 1^{3} 2_{+} ∗  3 3 2 3 2  2
S_{3} 8  2 3 2 2 3 2 2 2

Discriminant class: 29

∞ 11 13 29 a_{1} a_{2} a_{3} Over p = 11 Over p = 13  Over p = 29 2 3 5 7 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
4. 0_{+} 2_{+} 2_{+} 1_{+} 0 -143 -286 1^{3} 2_{+} ∗  1^{3} 2_{+} 1  1^{2} 1_{+} 29 29 2 2 3 3 2 2 3 2
S_{3} 3  2 2 2 2 3  2 3