Quartic Fields Unramified Outside of {∞, 29, 37}

Discriminant class: 1

∞ 29 37 a_{1} a_{2} a_{3}  Over p = 29  Over p = 37 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 31 41 43 47 53 59 61
1. 0_{+} 2_{−} 2_{−} 0 -33 0 2^{2} 2_{−} 1  V 2^{2} 2_{−} 1 V 22 22 22  22 22 22 22
V_{4}  4  22 22 22 22 22  22 22
2. 0_{+} 2_{+} 2_{+} -1 -15 4 1^{2} 1_{−} 29∗ 29∗ 1^{3} 2_{+} 1  3 3 3 3 22 3 3 3
A_{4} 2 45 1^{2} 1_{−} 29∗ 29∗ 22 22 3 22 22 3 3 3

Discriminant class: 29

∞ 29 37 a_{1} a_{2} a_{3}  Over p = 29  Over p = 37 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 31 41 43 47 53 59 61
3. 2_{−} 3_{−} 0_{+}  -1 4 -20 1^{4} 3_{−} 29 29 4 0_{+} ∗ ∗ 4 4 22  4 22 4 4
C_{4}  23  4 4 4 4  4
4. 2_{−} 3_{−} 2_{+}  -1 265 -803 1^{4} 3_{−} 29 29 2^{2} 2_{+} ∗ ∗ 4 4  4  4 4
C_{4} 90  3155  22 4 4 4 4  22 4

Discriminant class: 37

∞ 29 37  a_{1} a_{2} a_{3} Over p = 29  Over p = 37 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 31 41 43 47 53 59 61
5. 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
6. 2_{−} 2_{+} 3_{−}  1 264 266 2^{2} 2_{+} ∗ ∗ 1^{4} 3_{−} 37 37 4  4  4 4 4
C_{4} 162  18179  4 4  4 22  4 4

Discriminant class: 1073

∞ 29 37 a_{1} a_{2}  a_{3}  Over p = 29  Over p = 37  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 31 41 43 47 53 59 61
7. 0_{+} 3_{+} 3_{+}  1 -402 -1140 1^{4} 3_{+} 29∗ 29∗ 1^{4} 3_{+} 37∗ 37∗  4 4  4 4 22 22
C_{4} 2  3416  4  4 22 4  4
8. 0_{+} 3_{+} 3_{+}  -1 -402 -2079 1^{4} 3_{+} 29∗ 29∗ 1^{4} 3_{+} 37∗ 37∗ 22 4 4  4 4
C_{4} 2  197  4 22 4  4  4 22