Quartic Fields Unramified Outside of {∞, 3, 11}

Discriminant class: −11

∞ 3 11 a_{1} a_{2} a_{3} Over p = 3  Over p = 11  2 5 7 13 17 19 23 29
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
1. 1_{k} 2_{+} 3_{i} 2 -4 -5 2^{2} 2_{+} 1 V 1^{4} 3_{i} 11∗ 11 2 22 4 4 2 4 22 2
D_{4} -2 22 22 2 4 22 22 22 4
2. 1_{k} 4_{+} 3_{i} -1 -12 2 1^{3} 4_{+} 1  1^{4} 3_{i} 11∗ 11 4 3 4 2 2 2 22 4
S_{4} 37 3 3 2 2 3 3 3 4

Discriminant class: −3

∞ 3 11 a_{1} a_{2} a_{3}  Over p = 3  Over p = 11 2 5 7 13 17 19 23 29
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
3. 1_{k} 3_{i} 2_{+} -1 3 1 1^{3} 3_{i} 3∗  1^{3} 2_{+} ∗  4 4 3 3 4 22 4 2
S_{4}  -1  3 22 4 22 4 2 4 22

Discriminant class: 1

∞ 3 11 a_{1} a_{2} a_{3}  Over p = 3  Over p = 11 2 5 7 13 17 19 23 29
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
4. 2_{−} 2_{−} 2_{+} 1 -2 3 1^{2} 1_{i} 3∗ 3∗ 2^{2} 2_{+} 1 V 22 22 22 22 22 22 22 22
V_{4} 9 1^{2} 1_{i} 3∗ 3∗ 22 22 22 22 22 22

Discriminant class: 33

∞ 3 11 a_{1} a_{2} a_{3}  Over p = 3  Over p = 11  2 5 7 13 17 19 23 29
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d  s 31 37 41 43 47 53 59 61
5. 2_{−} 1_{k} 3_{k} -1 -1 2 1^{2} 1_{k} 3 3 1^{4} 3_{k} 11 11∗ 22 2 4 4 22 4 2 22
D_{4} 4 2 0_{+} ∗ ∗ 22 22 22 4 2 2 2 4
6. 2_{−} 5_{i} 1_{i} -1 0 1 1^{3} 5_{i} 3∗  1^{2} 1_{i} 11∗ 11∗ 3 4 4 2 3 4 4 3
S_{4}  2  2 0_{+} ∗  ∗ 22 3 3 4 4 2 2 4