Quartic Fields Unramified Outside of {∞, 13, 23}

Discriminant class: −299

∞ 13 23 a_{1} a_{2} a_{3}  Over p = 13  Over p = 23 2 3 5 7 11 17 19 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_{k} 2 8 7 1^{4} 3_{−} 13 13 1^{2} 1_{k} 23 23 4 22 22 22 22 2 22 22
D_{4} -17 2 0_{+} ∗ ∗ 4 22 4 2 4 2 4 2

Discriminant class: −23

∞ 13 23 a_{1} a_{2} a_{3}  Over p = 13  Over p = 23  2 3 5 7 11 17 19 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_{i} 1 0 -5 1^{2} 1_{+} 13 13 1^{2} 1_{i} 23∗ 23∗ 22 22 4 4 4 2 4 22
D_{4} -1 1^{2} 1_{+} 13 13 22 4 22 2 22 2 22 2

Discriminant class: 1

∞ 13 23 a_{1} a_{2} a_{3}  Over p = 13  Over p = 23  2 3 5 7 11 17 19 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 5 0 1^{2} 1_{+} 13 13 1^{2} 1_{i} 23∗ 23∗ 22  22 22 22 22 22
V_{4} 12 81 1^{2} 1_{+} 13 13 1^{2} 1_{i} 23∗ 23∗ 22 22 22 22 22 22 22 22

Discriminant class: 13

∞ 13 23 a_{1} a_{2} a_{3}  Over p = 13  Over p = 23  2 3 5 7 11 17 19 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_{−} 3_{−} 0_{+} -1 2 4 1^{4} 3_{−} 13 13 2 0_{+} ∗  ∗ 4  4 4 4 22 4
C_{4}  3  2 0_{+} ∗  ∗ 4 4 4 22 4  4
5. 0_{+} 3_{−} 2_{−} 1 -76 74 1^{4} 3_{−} 13 13 1^{2} 1_{k} 23 23 4  4 4 4  4
C_{4}  393  1^{2} 1_{k} 23 23 4 4 4  4 22 4 22
6. 2_{−} 1_{+} 2_{−} -2 4 -3 1^{2} 1_{+} 13 13 1^{2} 1_{i} 23∗ 23∗ 2 22 4 4 4 22 4 22
D_{4} 3 8 1^{2} 1_{i} 23∗ 23∗ 2 4 2 22 2 22 2 22
7. 2_{−} 3_{−} 2_{+} 0 13 0 1^{4} 3_{−} 13 13 2^{2} 2_{+} 1  V 4 22 2 2 2 22 2 22
D_{4} 8  117  4 2 4 22 4 22 4 22