Quartic Fields Unramified Outside of {∞, 19, 41}

Discriminant class: −19

∞ 19 41 a_{1} a_{2} a_{3}  Over p = 19  Over p = 41 2 3 5 7 11 13 17 23
G h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d s 29 31 37 43 47 53 59 61
1. 1_{k} 1_{i} 2_{+} -1 -1 -9 1^{2} 1_{i} 19∗ 19∗ 1^{3} 2_{+} ∗  4 2 3 22 22 4 3 3
S_{4}  9  4 4 2 3 3 2 2 3
2. 1_{k} 3_{i} 2_{+} 0 -38 -19 1^{4} 3_{i} 19∗ 19 1^{3} 2_{+} ∗  4 2 3  22 2 3 3
S_{4} 2  399  2 2 4 3 3 4 4 3
3. 1_{k} 3_{i} 2_{+} 2 11 86 1^{4} 3_{i} 19∗ 19 1^{3} 2_{+} ∗  2 2 3 22 22 4 3 3
S_{4} 2 -32 4 4 4 3 3 4 4 3

Discriminant class: 1

∞ 19 41 a_{1} a_{2} a_{3} Over p = 19  Over p = 41 2 3 5 7 11 13 17 23
G h a_{4} f^{e} c_{w} d s f^{e} c_{w} d s 29 31 37 43 47 53 59 61
4. 2_{−} 2_{+} 2_{−} 0 -11 0 2^{2} 2_{+} 1 V 2^{2} 2_{−} 1 V 22 22  22 22 22 22
V_{4} 5 225 22 22 22 22 22 22

Discriminant class: 41

∞ 19 41 a_{1} a_{2} a_{3} Over p = 19  Over p = 41 2 3 5 7 11 13 17 23
G h a_{4} f^{e} c_{w} d s f^{e} c_{w} d s 29 31 37 43 47 53 59 61
5. 0_{+} 0_{+} 3_{+}  -1 -15 -18 4 0_{+} ∗ ∗ 1^{4} 3_{+} 41 41 22 4 22 4 4 4 4
C_{4}  -4  4  22 4 4  22
6. 2_{−} 2_{−} 3_{+}  -1 190 392 2^{2} 2_{−} ∗ ∗ 1^{4} 3_{+} 41 41  4 22 4 4 4 4
C_{4} 82  4096  4 22 22 22 4 4 22 22