Low Degree Fields Ramifying Above {3, 41}

Discriminant class: −123

∞ 3 41 a_{1} a_{2} a_{3} Over p = 3  Over p = 41  2 5 7 11 13 17 19 23
G h  f^{e} c_{w} d s f^{e} c_{w} d  s 29 31 37 43 47 53 59 61
1. 1_{k} 1_{k} 1_{−} 0 123  1^{2} 1_{k} 3 3 1^{2} 1_{−} 41∗ 41∗ 2 2 2  2  2 2
C_{2} 2  2
2. 1_{k} 3_{k} 1_{−} 0 6 -3 1^{3} 3_{k} 3 1^{2} 1_{−} 41∗ 41∗ 2 2 2 3 2 2 2
S_{3} 2  3 3 3 3 3 3 2 3

Discriminant class: −3

∞ 3 41 a_{1} a_{2} a_{3} Over p = 3 Over p = 41 2 5 7 11 13 17 19 23
G h  f^{e} c_{w} d s f^{e} c_{w} d s 29 31 37 43 47 53 59 61
3. 1_{k} 3_{i} 2_{+} 0 0 -41 1^{3} 3_{i} 3∗  1^{3} 2_{+} ∗  2 2  2 3 2 3 2
S_{3}  2 3 3  2 2 2
4. 1_{k} 5_{i} 2_{+} 0 0 -369 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  2 2 3 2  2  2
S_{3} 4  2 3  3 2 2 2
5. 1_{k} 5_{i} 2_{+} 0 0 -123 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  2 2 3 2 3 2 3 2
S_{3} 2  2  3 3 2 2 2