Low Degree Fields Unramified Outside of {∞, 2, 53}

Discriminant class: −106

∞ 2 53 a_{1} a_{2} a_{3}  Over p = 2  Over p = 53  3 5 7 11 13 17 19 23
G h  f^{e} c_{w} d  s f^{e} c_{w} d  s 29 31 37 41 43 47 59 61
1. 1_{k} 3_{k} 1_{−} 0 106  1^{2} 3_{k} −2∗ −2∗ 1^{2} 1_{−} 53∗ 53∗ 2  2  2  2
C_{2} 6  2  2 2  2
2. 1_{k} 3_{k} 1_{−} -1 0 8 1^{2} 3_{k} −2∗ −2∗ 1^{2} 1_{−} 53∗ 53∗ 2 3 2 3 2 3 2 3
S_{3}  2 3 2 2 3 2 3

Discriminant class: −53

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

Discriminant class: −2

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

Discriminant class: −1

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

Discriminant class: 2

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

Discriminant class: 53

∞ 2 53 a_{1} a_{2} a_{3} Over p = 2  Over p = 53 3 5 7 11 13 17 19 23
G h  f^{e} c_{w} d s f^{e} c_{w} d s 29 31 37 41 43 47 59 61
9. 0_{+} 0_{+} 1_{+} 0 -53  2 0_{+} ∗ ∗ 1^{2} 1_{+} 53 53 2 2  2 2
C_{2}  2  2  2

Discriminant class: 106

∞ 2 53 a_{1} a_{2} a_{3}  Over p = 2  Over p = 53  3 5 7 11 13 17 19 23
G h  f^{e} c_{w} d s f^{e} c_{w} d  s 29 31 37 41 43 47 59 61
10. 0_{+} 3_{−} 1_{−} 0 -106  1^{2} 3_{−} 2∗ 2∗ 1^{2} 1_{−} 53∗ 53∗  2 2  2
C_{2} 2  2 2 2 2 2  2