Quartic Fields Ramifying Above {3, 13}

Discriminant class: −39

∞ 3 13 a_{1} a_{2} a_{3} Over p = 3  Over p = 13 2 5 7 11 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} 1_{k} 3_{−} -1 2 -9 1^{2} 1_{k} 3 3 1^{4} 3_{−} 13 13 22 22 4 22 2 4 2 2
D_{4} 3 2 0_{+} ∗ ∗ 4 4 22 22 22 2 22 22

Discriminant class: −3

∞ 3 13 a_{1} a_{2} a_{3}  Over p = 3  Over p = 13 2 5 7 11 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. 1_{k} 1_{k} 2_{−} 1 -1 1 1^{2} 1_{k} 3 3 2^{2} 2_{−} 1 V 4 4 22 4 2 22 2 2
D_{4}  1  2 0_{+} ∗ ∗  22 22 4  4 2 4 22
3. 1_{k} 5_{i} 2_{+} 0 -6 -5 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  4 4 3 4 4 22 2 4
S_{4}  -3  3 2 22 2 2 2 3

Discriminant class: 1

∞ 3 13 a_{1} a_{2} a_{3}  Over p = 3  Over p = 13 2 5 7 11 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
4. 2_{−} 2_{−} 2_{+} -1 4 3 1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{+} 13 13 22 22 22 22 22 22 22 22
V_{4} 2 9 1^{2} 1_{i} 3∗ 3∗ 1^{2} 1_{+} 13 13 22 22 22 22 22 22

Discriminant class: 13

∞ 3 13 a_{1} a_{2} a_{3}  Over p = 3  Over p = 13  2 5 7 11 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
5. 0_{+} 2_{−} 3_{−} -1 -11 -9 1^{2} 1_{i} 3∗ 3∗ 1^{4} 3_{−} 13 13 4 4 4 4  4  22
C_{4}  3  1^{2} 1_{i} 3∗ 3∗  4 4 4 22 4 22 4
6. 2_{−} 2_{+} 1_{−} -1 2 -2 2^{2} 2_{+} 1 V 1^{2} 1_{−} 13∗ 13∗ 4 4 2 4 22 2 22 22
D_{4} 1 2 0_{+} ∗ ∗ 2 2 4 4 22 4 22
7. 2_{−} 2_{+} 3_{−} 1 2 -4 2^{2} 2_{+} 1 V 1^{4} 3_{−} 13 13 2 2 4 2 22 4 22 22
D_{4} 4  16  4 4 2 22 2 22 2 22