Quartic Fields Ramifying Above {3, 17}

Discriminant class: −3

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

Discriminant class: 1

∞ 3 17 a_{1} a_{2} a_{3} Over p = 3  Over p = 17 2 5 7 11 13 19 23 29
G h a_{4} f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
3. 2_{−} 2_{+} 2_{−} -1 5 4 2^{2} 2_{+} 1 V 2^{2} 2_{−} 1 V 22 22 22 22  22 22
V_{4} 16 22 22 22 22 22 22 22
4. 2_{−} 4_{+} 2_{−} -1 6 -5 1^{3} 4_{+} 1  2^{2} 2_{−} 1 V 3 3 3 3 3 22 3 3
A_{4} 2 8 3 3 3 3 22 3 3

Discriminant class: 17

∞ 3 17 a_{1} a_{2} a_{3} Over p = 3  Over p = 17 2 5 7 11 13 19 23 29
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
5. 2_{−} 2_{−} 3_{+} 1 11 -1 2^{2} 2_{−} ∗ ∗ 1^{4} 3_{+} 17 17  4 4 4  22 4 4
C_{4} 10  52  4 4 4 22 22  4