Quartic Fields Unramified Outside of {∞, 37, 47}

Discriminant class: −1739

∞ 37 47 a_{1} a_{2} a_{3}  Over p = 37  Over p = 47 2 3 5 7 11 13 17 19
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d s 23 29 31 41 43 53 59 61
1. 1_{k} 3_{−} 1_{k} 1 5 -67 1^{4} 3_{−} 37 37 1^{2} 1_{k} 47 47 4  22  2 22 4 22
D_{4} 9  -99  2 0_{+} ∗ ∗ 22 22 22 2 22  4 4

Discriminant class: −47

∞ 37 47 a_{1} a_{2} a_{3} Over p = 37  Over p = 47 2 3 5 7 11 13 17 19
G h a_{4} f^{e} c_{w} d s f^{e} c_{w} d s 23 29 31 41 43 53 59 61
2. 1_{k} 2_{−} 1_{k} 2 -5 -6 2^{2} 2_{−} 1 V 1^{2} 1_{k} 47 47 22 22 4  2 4 22 4
D_{4} -28 2 0_{+} ∗ ∗ 4 4 4 2 4 22 22

Discriminant class: 1

∞ 37 47 a_{1} a_{2} a_{3}  Over p = 37  Over p = 47  2 3 5 7 11 13 17 19
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d  s 23 29 31 41 43 53 59 61
3. 2_{−} 2_{+} 2_{−} 0 5 0 1^{2} 1_{+} 37 37 1^{2} 1_{i} 47∗ 47∗ 22  22  22 22 22 22
V_{4} 50  441  1^{2} 1_{+} 37 37 1^{2} 1_{i} 47∗ 47∗ 22 22 22 22 22  22 22
4. 2_{−} 2_{+} 2_{−} 0 5 -47 1^{3} 2_{+} 1  1^{2} 1_{k} 47 47 3 3 3 3  3 3 3
A_{4} 4  112  1^{2} 1_{k} 47 47 22 22 22 3 22 3 3 3

Discriminant class: 37

∞ 37 47  a_{1} a_{2} a_{3}  Over p = 37  Over p = 47  2 3 5 7 11 13 17 19
G  h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d  s 23 29 31 41 43 53 59 61
5. 2_{−} 3_{−} 0_{+}  -1  5 -7 1^{4} 3_{−} 37 37  4 22 4  22 4 4 4
C_{4}  49  4 4 4 22 4  4 4
6. 0_{+} 3_{−} 2_{−}  -1 -439 437 1^{4} 3_{−} 37 37 1^{2} 1_{i} 47∗ 47∗ 4 22 4  4 4 4
C_{4} 5  44893  1^{2} 1_{i} 47∗ 47∗ 4 4 4  4  4 4
7. 2_{−} 1_{−} 2_{+}  -1  3 -19 1^{2} 1_{−} 37∗ 37∗ 2^{2} 2_{+} 1  V 2 22 4  22 4 2 4
D_{4} 5  32  2 0_{+} ∗  ∗  4 4 4 22 4  2 2
8. 2_{−} 3_{−} 2_{+}  2 20 19 1^{4} 3_{−} 37 37 2^{2} 2_{+} 1  V 4  2  22 2 4 2
D_{4} 180  525  2 2 2 22 2  4 4