Quartic Fields Unramified Outside of {∞, 29, 31}

Discriminant class: −31

∞ 29 31 a_{1} a_{2} a_{3}  Over p = 29  Over p = 31  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 37 41 43 47 53 59 61
1. 1_{k} 2_{−} 3_{k}  -1 12 -343 1^{2} 1_{+} 29 29 1^{4} 3_{k} 31∗ 31 3 2 3 3 2 4 4 3
S_{4} 6 -120 1^{2} 1_{−} 29∗ 29∗ 2 2 3 4 4 3 4

Discriminant class: 1

∞ 29 31 a_{1} a_{2} a_{3} Over p = 29  Over p = 31 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 37 41 43 47 53 59 61
2. 2_{−} 2_{−} 2_{+} 0 1 0 2^{2} 2_{−} 1 V 2^{2} 2_{+} 1 V 22 22  22 22 22 22
V_{4} 21 225 22 22 22 22 22 22 22
3. 2_{−} 2_{−} 2_{+} -1 17 -12 2^{2} 2_{−} 1 V 1^{3} 2_{+} 1  22 3 3 3 3 3 3 3
A_{4} 2 28 22 3 3 3 3 3 22

Discriminant class: 29

∞ 29 31 a_{1} a_{2} a_{3}  Over p = 29  Over p = 31 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 37 41 43 47 53 59 61
4. 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
5. 0_{+} 3_{−} 2_{−}  1 -228 -676 1^{4} 3_{−} 29 29 2^{2} 2_{−} ∗ ∗ 4 4 22  4  4 4
C_{4}  1183  22 4 4 4 4 22  4