Quartic Fields Unramified Outside of {∞, 41, 47}

Discriminant class: −1927

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

Discriminant class: 1

∞ 41 47 a_{1} a_{2} a_{3} Over p = 41  Over p = 47 2 3 5 7 11 13 17 19
G h a_{4} f^{e} c_{w} d s f^{e} c_{w} d s 23 29 31 37 43 53 59 61
2. 2_{−} 2_{−} 2_{+} 0 3 0 2^{2} 2_{−} 1 V 2^{2} 2_{+} 1 V  22 22 22 22 22 22 22
V_{4} 45 484 22 22 22 22 22

Discriminant class: 41

∞ 41 47  a_{1} a_{2} a_{3}  Over p = 41  Over p = 47 2 3 5 7 11 13 17 19
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d s 23 29 31 37 43 53 59 61
3. 0_{+} 3_{+} 0_{+}  -1 -15 -18 1^{4} 3_{+} 41 41 4 0_{+} ∗ ∗ 22 4 22 4 4 4 4 4
C_{4}  -4  4  22 4  22
4. 2_{−} 3_{+} 2_{−}  -1 477 966 1^{4} 3_{+} 41 41 2^{2} 2_{−} ∗ ∗ 22 4  4 4 4 4 4
C_{4} 50  23612  22 4 22  4  22