Quartic Fields Ramifying Above {5, 13}

Discriminant class: 1

∞ 5 13 a_{1} a_{2} a_{3} Over p = 5  Over p = 13 2 3 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. 0_{+} 2_{−} 2_{−} 0 -9 0 2^{2} 2_{−} 1 V 2^{2} 2_{−} 1 V 22 22 22 22 22 22 22
V_{4} 4 22 22 22 22 22 22 22
2. 2_{−} 2_{−} 2_{+} -1 0 -3 2^{2} 2_{−} 1 V 1^{3} 2_{+} 1  3 3 3 3 3 3 3 3
A_{4} 2 4 22 3 3 3 22 22 3 3

Discriminant class: 5

∞ 5 13 a_{1} a_{2} a_{3} Over p = 5  Over p = 13 2 3 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
3. 2_{−} 3_{−} 2_{+} -1 16 -16 1^{4} 3_{−} 5 5 2^{2} 2_{+} ∗ ∗ 4 4 4 22 4  4 22
C_{4} 2 61 22 4 22 4 4 4

Discriminant class: 13

∞ 5 13 a_{1} a_{2} a_{3} Over p = 5  Over p = 13 2 3 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_{+} 3_{−} 1 15 -17 2^{2} 2_{+} ∗ ∗ 1^{4} 3_{−} 13 13 4 22 4 4  4
C_{4} 2 29 4 4 4 4 22 4

Discriminant class: 65

∞ 5 13 a_{1} a_{2} a_{3}  Over p = 5  Over p = 13  2 3 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_{+} 3_{+} 3_{+} -1 -24 4 1^{4} 3_{+} 5∗ 5∗ 1^{4} 3_{+} 13∗ 13∗  4 22 4 4 4 4 22
C_{4} 2 16 4 22 4 4 22 4 4
6. 0_{+} 3_{+} 3_{+} 1 -24 -69 1^{4} 3_{+} 5∗ 5∗ 1^{4} 3_{+} 13∗ 13∗ 22 4  4 4 4 4 22
C_{4} 2  -49  4  4 4  4 4