Quartic Fields Unramified Outside of {∞, 19, 29}

Discriminant class: −19

∞ 19 29 a_{1} a_{2} a_{3}  Over p = 19  Over p = 29 2 3 5 7 11 13 17 23
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_{+} -2 -46 123 1^{4} 3_{i} 19∗ 19 1^{3} 2_{+} ∗  4 2 3 3 22 2 22 22
S_{4} 405 4 2 2 3 3 2 2 3

Discriminant class: 1

∞ 19 29 a_{1} a_{2} a_{3} Over p = 19  Over p = 29 2 3 5 7 11 13 17 23
G h a_{4} f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
2. 2_{−} 2_{+} 2_{−} 0 -5 0 2^{2} 2_{+} 1 V 2^{2} 2_{−} 1 V 22 22  22 22 22
V_{4} 13 144 22 22 22 22 22 22 22 22

Discriminant class: 29

∞ 19 29 a_{1} a_{2} a_{3} Over p = 19  Over p = 29 2 3 5 7 11 13 17 23
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_{−} 0_{+} 3_{−} -1  4 -20 4 0_{+} ∗ ∗ 1^{4} 3_{−} 29 29 4 4 22  4 22 4
C_{4}  23  4 4 4 4 4  4
4. 0_{+} 2_{−} 3_{−} 1 -141 -415 2^{2} 2_{−} ∗ ∗ 1^{4} 3_{−} 29 29 4 4 22  4  4
C_{4} 5  313  4 4 4 4 4 22 22 4