Quartic Fields Unramified Outside of {∞, 7, 37}

Discriminant class: −259

∞ 7 37 a_{1} a_{2} a_{3} Over p = 7  Over p = 37 2 3 5 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 81 1^{2} 1_{k} 7 7 1^{4} 3_{−} 37 37 4 2 22  22 22 22 4
D_{4} -99 2 0_{+} ∗ ∗ 4 22 2 4 2 22 22 22

Discriminant class: −7

∞ 7 37 a_{1} a_{2} a_{3} Over p = 7  Over p = 37 2 3 5 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_{k} 2_{−} 0 -3 0 1^{2} 1_{k} 7 7 2^{2} 2_{−} 1 V 22 2 4 22 4 4 4 22
D_{4} -7 2 0_{+} ∗ ∗ 22 4 2 22 2 22 4 4

Discriminant class: 1

∞ 7 37 a_{1} a_{2} a_{3}  Over p = 7  Over p = 37 2 3 5 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
3. 2_{−} 2_{−} 2_{+} 0 -15 0 1^{2} 1_{i} 7∗ 7∗ 1^{2} 1_{+} 37 37 22 22 22  22 22 22 22
V_{4} 2 121 1^{2} 1_{i} 7∗ 7∗ 1^{2} 1_{+} 37 37 22 22 22 22 22 22 22

Discriminant class: 37

∞ 7 37 a_{1} a_{2} a_{3}  Over p = 7  Over p = 37  2 3 5 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_{−} 0_{+} 3_{−} -1 5 -7  1^{4} 3_{−} 37 37 4 22 4 22 4 4 4 4
C_{4} 49 4 4 22 4 4 4
5. 0_{+} 2_{−} 3_{−} 1 -69 -67 1^{2} 1_{i} 7∗ 7∗ 1^{4} 3_{−} 37 37 4  4 22 4 4 4 4
C_{4} 863 1^{2} 1_{i} 7∗ 7∗ 4 4 4 22 4 4
6. 2_{−} 2_{+} 1_{−} -2 0  1 2^{2} 2_{+} 1 V 1^{2} 1_{−} 37∗ 37∗ 2 22 4 22 4 4 4 2
D_{4} 2 2 0_{+} ∗ ∗ 2 4 22 2 22 22 4 4
7. 2_{−} 2_{+} 3_{−} -1 42 -118 2^{2} 2_{+} 1 V 1^{4} 3_{−} 37 37 4 22 2  2 2 2 4
D_{4} 4  197  4 2 22 4 22 22 2 2