Quartic Fields Unramified Outside of {∞, 17, 37}

Discriminant class: 1

∞ 17 37 a_{1} a_{2} a_{3} Over p = 17  Over p = 37 2 3 5 7 11 13 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 -27 0 2^{2} 2_{−} 1 V 2^{2} 2_{−} 1 V 22 22 22 22 22 22 22 22
V_{4} 25 22 22 22 22 22 22

Discriminant class: 17

∞ 17 37 a_{1} a_{2} a_{3}  Over p = 17  Over p = 37 2 3 5 7 11 13 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. 0_{+} 3_{+} 0_{+}  1 -6 -1 1^{4} 3_{+} 17 17 4 0_{+} ∗ ∗ 22 4 4 4 4  22 4
C_{4}  1  4 4 4 22  22 22 4
3. 0_{+} 3_{+} 2_{+}  -1 -159 1 1^{4} 3_{+} 17 17 2^{2} 2_{+} ∗ ∗  4 4 4 4 22  4
C_{4} 10  5662  4 4 4  22  4

Discriminant class: 37

∞ 17 37 a_{1} a_{2} a_{3} Over p = 17  Over p = 37 2 3 5 7 11 13 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 4  22 4 4 4
C_{4}  49  4 4 22 4  4 4
5. 2_{−} 2_{+} 3_{−}  -1 153 -155 2^{2} 2_{+} ∗ ∗ 1^{4} 3_{−} 37 37 4  4 22  4 4 4
C_{4} 58  6413  4 4  4  4 4

Discriminant class: 629

∞ 17 37 a_{1} a_{2}  a_{3}  Over p = 17  Over p = 37  2 3 5 7 11 13 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_{−} 3_{−} 3_{+}  1 79 2005 1^{4} 3_{−} 17∗ 17∗ 1^{4} 3_{+} 37∗ 37∗ 4 4 22 4 4 4 4
C_{4} 36  8263  4 4  22 4 22
7. 2_{−} 3_{−} 3_{+}  1 79 -1769 1^{4} 3_{−} 17∗ 17∗ 1^{4} 3_{+} 37∗ 37∗ 4 4  4 4 4 4 22
C_{4} 52  8263  22 22 4 4  22 4