Quartic Fields Unramified Outside of {∞, 5, 37}

Discriminant class: 1

∞ 5 37 a_{1} a_{2} a_{3} Over p = 5  Over p = 37 2 3 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. 0_{+} 2_{−} 2_{−} 0 -21 0 2^{2} 2_{−} 1 V 2^{2} 2_{−} 1 V 22 22 22  22 22 22 22
V_{4} 64 22 22 22 22 22 22 22

Discriminant class: 5

∞ 5 37 a_{1} a_{2} a_{3} Over p = 5  Over p = 37 2 3 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. 2_{−} 3_{−} 0_{+} -1 1 -1 1^{4} 3_{−} 5 5 4 0_{+} ∗ ∗ 4 4 4  4 4 22 4
C_{4}  1  22  4 4 4 22
3. 2_{−} 3_{−} 2_{+} -1 46 -46 1^{4} 3_{−} 5 5 2^{2} 2_{+} ∗ ∗ 4 4 4  4 4  4
C_{4} 10  451  22  4 4 4  22

Discriminant class: 37

∞ 5 37 a_{1} a_{2} a_{3} Over p = 5  Over p = 37 2 3 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_{−} 0_{+} 3_{−} -1 5 -7 4 0_{+} ∗ ∗ 1^{4} 3_{−} 37 37 4 22  22 4 4 4 4
C_{4} 49 4 4 22 4 4 4
5. 2_{−} 2_{+} 3_{−} -1 42 -44 2^{2} 2_{+} ∗ ∗ 1^{4} 3_{−} 37 37 4  22 22 4 4 4 4
C_{4} 10 641 4 4 22 4 22 22 4 4

Discriminant class: 185

∞ 5 37 a_{1} a_{2} a_{3}  Over p = 5  Over p = 37  2 3 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. 0_{+} 3_{+} 3_{+} -1 -69 -266 1^{4} 3_{+} 5∗ 5∗ 1^{4} 3_{+} 37∗ 37∗ 22 4 4 22 22  4 22
C_{4} 2  -284  4 4 22 22 4 4 4 4
7. 0_{+} 3_{+} 3_{+}  1 -69 -289 1^{4} 3_{+} 5∗ 5∗ 1^{4} 3_{+} 37∗ 37∗  4 4 22  22 4
C_{4} 2  -284  4 4 22  4 4 4 4