Quartic Fields Unramified Outside of {∞, 3, 37}

Discriminant class: −111

∞ 3 37 a_{1} a_{2} a_{3} Over p = 3  Over p = 37 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 41 43 47 53 59 61
1. 1_{k} 1_{k} 3_{−} 1 5 -30 1^{2} 1_{k} 3 3 1^{4} 3_{−} 37 37 22 22 22 2 4 22 4 22
D_{4} 12 2 0_{+} ∗ ∗ 22 4 2 4 2 2 22 4

Discriminant class: −3

∞ 3 37 a_{1} a_{2} a_{3}  Over p = 3  Over p = 37 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 41 43 47 53 59 61
2. 1_{k} 1_{i} 2_{+} 0 5 0 1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{+} 37 37 4 4 22 2 22 4 22 4
D_{4} -3 1^{2} 1_{+} 37 37 4 22 2 22 2 2 4 22
3. 1_{k} 3_{i} 2_{+} -1 3 -7 1^{4} 3_{i} 3∗ 3 1^{3} 2_{+} 1  4 4 3 4 3 2 22 4
S_{4} 1 2 3 2 3 2 4 4

Discriminant class: 1

∞ 3 37 a_{1} a_{2} a_{3}  Over p = 3  Over p = 37 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 41 43 47 53 59 61
4. 2_{−} 2_{−} 2_{+} -1 10 9 1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{+} 37 37 22 22  22 22 22 22 22
V_{4} 4  81  1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{+} 37 37 22 22 22 22 22 22 22 22
5. 2_{−} 4_{+} 2_{−} -1 6 11 1^{3} 4_{+} 1  2^{2} 2_{−} 1 V 3 3 3 3 3 22  3
A_{4} 2  10  3 3 3 3 3 22 3 3

Discriminant class: 37

∞ 3 37 a_{1} a_{2} a_{3}  Over p = 3  Over p = 37 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 41 43 47 53 59 61
6. 2_{−} 0_{+} 3_{−} -1 5 -7 2 0_{+} ∗ ∗ 1^{4} 3_{−} 37 37 4 4  22 4 4 4 4
C_{4} 49 2 0_{+} ∗ ∗ 4 4 22 4 4 4
7. 0_{+} 2_{−} 3_{−} 1 -32 -30 1^{2} 1_{k} 3 3 1^{4} 3_{−} 37 37 4 4  4 4 4 4
C_{4} 123 1^{2} 1_{k} 3 3 4 4 4 22 22 4 4
8. 2_{−} 2_{−} 1_{+} -2 4 -3 1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{+} 37 37 4 4 22 22 2 4 2 4
D_{4} 3 1^{2} 1_{i} 3∗ 3∗ 4 2 22 2 22 22 4 2
9. 2_{−} 2_{+} 3_{−} 2 20 19 2^{2} 2_{+} 1 V 1^{4} 3_{−} 37 37 2 2 22 22 4 2 4 2
D_{4} 8  118  2 4 22 4 22 22 2 4