Quartic Fields Unramified Outside of {∞, 3, 47}

Discriminant class: −47

∞ 3 47 a_{1} a_{2} a_{3}  Over p = 3  Over p = 47  2 5 7 11 13 17 19 23
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d s 29 31 37 41 43 53 59 61
1. 1_{k} 2_{−} 3_{k} -2 25 -24 1^{2} 1_{i} 3∗ 3∗ 1^{4} 3_{k} 47∗ 47 22 2 22 2 4 22 4 2
D_{4} 3 1^{2} 1_{i} 3∗ 3∗ 2 4 2 4 22 22

Discriminant class: −3

∞ 3 47 a_{1} a_{2} a_{3}  Over p = 3  Over p = 47 2 5 7 11 13 17 19 23
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d s 29 31 37 41 43 53 59 61
2. 1_{k} 3_{i} 2_{+} -1 -6 -8 1^{3} 3_{i} 3∗  1^{3} 2_{+} ∗  4 2 3 2 22 2 3 4
S_{4}  -1  4 22 22 4 22 4 2 3
3. 1_{k} 5_{i} 2_{+} -1 9 -35 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  2 4 22 2 3 4 22 2
S_{4}  2  4 3 3 4 3 4 4 3
4. 1_{k} 5_{i} 2_{+} -1 -9 -15 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  4 2 22 4 3 2 22 2
S_{4}  -3  4 3 3 2 3 2 2 3
5. 1_{k} 5_{i} 2_{+} -1 0 12 1^{3} 5_{i} 3∗  1^{3} 2_{+} ∗  4 4 22 4 3 4 22 2
S_{4}  -3  2 3 3 4 3 4 4 3

Discriminant class: 1

∞ 3 47 a_{1} a_{2} a_{3}  Over p = 3  Over p = 47 2 5 7 11 13 17 19 23
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d s 29 31 37 41 43 53 59 61
6. 2_{−} 2_{−} 2_{+} -1 -11 -12 1^{2} 1_{i} 3∗ 3∗ 2^{2} 2_{+} 1 V 22 22  22 22 22 22 22
V_{4} 5 144 1^{2} 1_{i} 3∗ 3∗ 22 22 22 22 22 22

Discriminant class: 141

∞ 3 47 a_{1} a_{2} a_{3}  Over p = 3  Over p = 47  2 5 7 11 13 17 19 23
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d  s 29 31 37 41 43 53 59 61
7. 2_{−} 1_{i} 3_{i} -1 18 -3 1^{2} 1_{i} 3∗ 3∗ 1^{4} 3_{i} 47 47∗ 2 22 22 22 4 2 4 22
D_{4} 5 9 22 4 22 4 2 2
8. 2_{−} 5_{i} 1_{i} 2 0 -3 1^{3} 5_{i} 3∗  1^{2} 1_{i} 47∗ 47∗ 4 3 22 3 2 4 4 3
S_{4}  3  2 0_{+} ∗  ∗  2 3 3 2 4 2 3