Quartic Fields Unramified Outside of {∞, 17, 19}

Discriminant class: −323

∞ 17 19 a_{1} a_{2} a_{3}  Over p = 17  Over p = 19  2 3 5 7 11 13 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_{+} 1_{i} -2 10 -9 1^{4} 3_{+} 17 17 1^{2} 1_{i} 19∗ 19∗ 2 22 4 4 4 2 4 22
D_{4} 2 -18 22 22 22 22 2 2 4

Discriminant class: −19

∞ 17 19 a_{1} a_{2} a_{3} Over p = 17  Over p = 19 2 3 5 7 11 13 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. 1_{k} 2_{−} 1_{k} -2 -2 3 2^{2} 2_{−} 1 V 1^{2} 1_{k} 19 19 2 4 22 22 22 2 22 4
D_{4} -2 2 0_{+} ∗ ∗ 4 4 4 22 22 2 2 22

Discriminant class: 1

∞ 17 19 a_{1} a_{2} a_{3}  Over p = 17  Over p = 19  2 3 5 7 11 13 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_{−} 2_{+} 2_{−} 0 1 0 1^{2} 1_{+} 17 17 1^{2} 1_{i} 19∗ 19∗ 22 22 22 22 22 22 22 22
V_{4} 2 81 1^{2} 1_{+} 17 17 1^{2} 1_{i} 19∗ 19∗ 22 22 22 22 22 22

Discriminant class: 17

∞ 17 19 a_{1} a_{2} a_{3}  Over p = 17  Over p = 19  2 3 5 7 11 13 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. 0_{+} 3_{+} 0_{+}  1 -6 -1 1^{4} 3_{+} 17 17 2 0_{+} ∗  ∗ 22 4 4 4 4  4 4
C_{4}  1  2 0_{+} ∗  ∗ 4 4 4 22  22 22 4
5. 2_{−} 3_{+} 2_{−}  1 79 -1 1^{4} 3_{+} 17 17 1^{2} 1_{k} 19 19  4 4 4 4 22 4 4
C_{4} 52  1616  1^{2} 1_{k} 19 19 4 4 4 22  4
6. 2_{−} 1_{−} 2_{+}  -1 1 2 1^{2} 1_{−} 17∗ 17∗ 2^{2} 2_{+} 1  V 22 4 2 2 2 22 2 4
D_{4}  4  2 0_{+} ∗  ∗  4 4 4 22 22 22 22 2
7. 2_{−} 3_{+} 2_{−}  0 17 0 1^{4} 3_{+} 17 17 1^{2} 1_{i} 19∗ 19∗ 22 2 4 4 4 22 4 2
D_{4} 8  153  1^{2} 1_{i} 19∗ 19∗ 2 2 2  22 22 22 4