Quartic Fields Ramifying Above {3, 7}

Discriminant class: −3

∞ 3 7 a_{1} a_{2} a_{3}  Over p = 3  Over p = 7 2 5 11 13 17 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 -3 -1 1^{4} 3_{i} 3∗ 3 2^{2} 2_{+} 1 V 4 2 4 22 2 22 4 4
D_{4}  1  22 22 2 22 2 4 2 22
2. 1_{k} 5_{i} 2_{+} -1 -3 -7 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  4 4 4 3 2 3 2 4
S_{4}  1  22 22 4 3 2 2 2 3

Discriminant class: 1

∞ 3 7 a_{1} a_{2} a_{3} Over p = 3  Over p = 7  2 5 11 13 17 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 -1 2 2^{2} 2_{+} 1 V 1^{2} 1_{i} 7∗ 7∗ 22 22 22 22 22 22 22 22
V_{4} 4 1^{2} 1_{i} 7∗ 7∗ 22 22 22 22 22 22

Discriminant class: 21

∞ 3 7 a_{1} a_{2} a_{3}  Over p = 3  Over p = 7 2 5 11 13 17 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
4. 2_{−} 3_{k} 1_{k} 1 0 -2 1^{4} 3_{k} 3 3∗ 1^{2} 1_{k} 7 7 4 22 4 2 22 2 4 4
D_{4}  1  2 0_{+} ∗ ∗ 2 22 22 22 22 4 22 2