Quartic Fields Unramified Outside of {∞, 3, 5}

Discriminant class: −3

∞ 3 5 a_{1} a_{2} a_{3}  Over p = 3  Over p = 5 2 7 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} 5_{i} 2_{+} -1 0 4 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  4 22 4 3 4 3 2 4
S_{4}  -1  22 3 4 3 4 4 4 3

Discriminant class: 1

∞ 3 5 a_{1} a_{2} a_{3} Over p = 3  Over p = 5 2 7 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
2. 2_{−} 2_{+} 2_{−} -1 2 1 2^{2} 2_{+} 1 V 2^{2} 2_{−} 1 V 22 22 22 22 22  22 22
V_{4} 1 22 22 22 22 22 22

Discriminant class: 5

∞ 3 5 a_{1} a_{2} a_{3} Over p = 3  Over p = 5 2 7 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_{−} 0_{+} 3_{−} -1 1 -1 4 0_{+} ∗ ∗ 1^{4} 3_{−} 5 5 4 4  4 4 22 4 22
C_{4} 1 4 4 4 4 22
4. 0_{+} 2_{−} 3_{−} -1 -4 4 2^{2} 2_{−} ∗ ∗ 1^{4} 3_{−} 5 5 4 4 22 4 4 22 4
C_{4}  1  4 22 4 4 4