Quartic Fields Unramified Outside of {∞, 2}

Discriminant class: −2

∞  2 a_{1} a_{2} a_{3}  Over p = 2  3 5 7 11 13 17 19 23
G  h a_{4}  f^{e} c_{w} d s 29 31 37 41 43 47 53 59
1. 1_{k} 11_{i} 0 0 0 1^{4} 11_{i} −2 2 22 4 2 22 4 22 22 2
D_{4} -2 4 2 4 22 22 2 4 22

Discriminant class: −1

∞  2 a_{1} a_{2} a_{3}  Over p = 2  3 5 7 11 13 17 19 23
G  h a_{4}  f^{e} c_{w} d s 29 31 37 41 43 47 53 59
2. 1_{k} 10_{i} 0 -2 0 1^{4} 10_{i} −1 2 4 22 2 4 22 22 4 2
D_{4} -1 22 2 22 4 2 22 4

Discriminant class: 1

∞ 2 a_{1} a_{2} a_{3}  Over p = 2 3 5 7 11 13 17 19 23
G h a_{4}  f^{e} c_{w} d s 29 31 37 41 43 47 53 59
3. 2_{−} 8_{−} 0 0 0 1^{4} 8_{−} 1 {−2 \atop −1} 22 22 22 22 22  22 22
V_{4} 1 22 22 22 22 22 22 22

Discriminant class: 2

∞  2 a_{1} a_{2} a_{3}  Over p = 2  3 5 7 11 13 17 19 23
G  h a_{4}  f^{e} c_{w} d s 29 31 37 41 43 47 53 59
4. 2_{−} 11_{−} 0 4 0 1^{4} 11_{−} 2 2 4 4  4 4  4
C_{4}  2  4 22 4 22 4 22 4 4
5. 0_{+} 11_{+} 0 -4 0 1^{4} 11_{+} 2 2 4 4 22 4 4  4 22
C_{4} 2 4 4 22 4 4 4
6. 2_{−} 9_{−} 0 -2 0 1^{4} 9_{−} 2 −1 4 2 22 4 2 22 4 22
D_{4} 2 2 22 2 4 22 2 4
7. 2_{−} 11_{−} 0 0 0 1^{4} 11_{−} 2 −2 2 4 22 2 4 22 2 22
D_{4}  2  4 22 4 22 2 22 4 2