Quartic Fields Ramifying Above {13, 37}

Discriminant class: 1

∞ 13 37 a_{1} a_{2} a_{3} Over p = 13  Over p = 37 2 3 5 7 11 17 19 23
G h a_{4} f^{e} c_{w} d s f^{e} c_{w} d s 29 31 41 43 47 53 59 61
1. 0_{+} 2_{−} 2_{−} 0 -25 0 2^{2} 2_{−} 1 V 2^{2} 2_{−} 1 V 22  22 22 22 22 22 22
V_{4} 36 22 22 22 22 22 22 22

Discriminant class: 13

∞ 13 37 a_{1} a_{2} a_{3}  Over p = 13  Over p = 37 2 3 5 7 11 17 19 23
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d s 29 31 41 43 47 53 59 61
2. 2_{−} 3_{−} 2_{+}  -1 119 121 1^{4} 3_{−} 13 13 2^{2} 2_{+} ∗ ∗ 4  4 4 4  4
C_{4} 50  1173  22 4 4  4  4 22

Discriminant class: 37

∞ 13 37 a_{1} a_{2} a_{3} Over p = 13  Over p = 37 2 3 5 7 11 17 19 23
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d s 29 31 41 43 47 53 59 61
3. 2_{−} 2_{+} 3_{−}  1 116 118 2^{2} 2_{+} ∗ ∗ 1^{4} 3_{−} 37 37 4 22 4 22  4 4 4
C_{4} 10  3823  4 4  4 22  4 4

Discriminant class: 481

∞ 13 37  a_{1} a_{2} a_{3}  Over p = 13  Over p = 37  2 3 5 7 11 17 19 23
G h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d  s 29 31 41 43 47 53 59 61
4. 0_{+} 3_{+} 3_{+}  1 -180 451 1^{4} 3_{+} 13∗ 13∗ 1^{4} 3_{+} 37∗ 37∗ 22 22  4 4 4  4
C_{4} 2  419  4 22 4 4 4  22 4
5. 0_{+} 3_{+} 3_{+}  1 -180 -992 1^{4} 3_{+} 13∗ 13∗ 1^{4} 3_{+} 37∗ 37∗  22 22 4 4 4 22 4
C_{4} 2  -1024  4  4 4 4  4