Quartic Fields Unramified Outside of {∞, 7, 11}

Discriminant class: −7

∞ 7 11 a_{1} a_{2} a_{3}  Over p = 7  Over p = 11  2 3 5 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} 3_{k} 2_{−} 0 -7 0 1^{4} 3_{k} 7∗ 7 1^{2} 1_{i} 11∗ 11∗ 22 4 4 2 2 2 22 22
D_{4} -7 1^{2} 1_{i} 11∗ 11∗ 4 2 22 4 22 4 2

Discriminant class: 1

∞ 7 11 a_{1} a_{2} a_{3} Over p = 7  Over p = 11  2 3 5 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
2. 2_{−} 2_{+} 2_{−} 0 9 0 2^{2} 2_{+} 1 V 1^{2} 1_{i} 11∗ 11∗ 22 22 22 22 22 22  22
V_{4} 1 1^{2} 1_{i} 11∗ 11∗ 22 22 22 22 22 22

Discriminant class: 77

∞ 7 11 a_{1} a_{2} a_{3}  Over p = 7  Over p = 11  2 3 5 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. 2_{−} 3_{i} 1_{i} 2 -2 -3 1^{4} 3_{i} 7 7∗ 1^{2} 1_{i} 11∗ 11∗ 2 4 4 22 22 22 22 2
D_{4}  4  4  22 2 4 22 4 22