Low Degree Fields Unramified Outside of {∞, 31, 53}

Discriminant class: −1643

∞ 31 53 a_{1} a_{2} a_{3}  Over p = 31  Over p = 53  2 3 5 7 11 13 17 19
G h  f^{e} c_{w} d s f^{e} c_{w} d  s 23 29 37 41 43 47 59 61
1. 1_{k} 1_{k} 1_{−} 0 1643  1^{2} 1_{k} 31 31 1^{2} 1_{−} 53∗ 53∗ 2  2  2 2 2 2
C_{2} 10  2 2 2 2

Discriminant class: −31

∞ 31 53 a_{1} a_{2} a_{3}  Over p = 31  Over p = 53 2 3 5 7 11 13 17 19
G h  f^{e} c_{w} d  s f^{e} c_{w} d s 23 29 37 41 43 47 59 61
2. 1_{k} 1_{i} 0_{+} 0 31  1^{2} 1_{i} 31∗ 31∗ 2 0_{+} ∗ ∗  2  2 2 2
C_{2} 3  2 2 2  2  2
3. 1_{k} 1_{i} 0_{+} 0 1 -1 1^{2} 1_{i} 31∗ 31∗ 2 0_{+} ∗ ∗ 3 2 3 3 2 2 2 3
S_{3} 2 2 2 3 2 3 2
4. 1_{k} 1_{i} 2_{+} 0 -53 -159 1^{2} 1_{i} 31∗ 31∗ 1^{3} 2_{+} ∗  3 2 3 3 2 2 2
S_{3} 3  2 2 2 3 2 3 3 2
5. 1_{k} 1_{i} 2_{+} -1 18 -55 1^{2} 1_{i} 31∗ 31∗ 1^{3} 2_{+} ∗  3 2  3 2 2 2 3
S_{3} 3  2 2 2  2 3  2
6. 1_{k} 1_{i} 2_{+} -1 18 104 1^{2} 1_{i} 31∗ 31∗ 1^{3} 2_{+} ∗  2 3  2 2 2 3
S_{3} 3  2 2 2 3 2 3 3 2

Discriminant class: 1

∞ 31 53 a_{1} a_{2} a_{3} Over p = 31 Over p = 53 2 3 5 7 11 13 17 19
G h  f^{e} c_{w} d s f^{e} c_{w} d s 23 29 37 41 43 47 59 61
7. 0_{+} 2_{+} 0_{+} 1 -10 -8 1^{3} 2_{+} 1  3 0_{+} 1  3 3 3 3 3 3 3
C_{3}  3 3 3  3

Discriminant class: 53

∞ 31 53 a_{1} a_{2} a_{3} Over p = 31  Over p = 53 2 3 5 7 11 13 17 19
G h  f^{e} c_{w} d s f^{e} c_{w} d s 23 29 37 41 43 47 59 61
8. 0_{+} 0_{+} 1_{+} 0 -53  2 0_{+} ∗ ∗ 1^{2} 1_{+} 53 53 2 2 2  2
C_{2}  2  2  2