Quartic Fields Ramifying Above {19, 37}

Discriminant class: 1

∞ 19 37 a_{1} a_{2} a_{3} Over p = 19  Over p = 37 2 3 5 7 11 13 17 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. 2_{−} 2_{+} 2_{−} 0 -9 0 2^{2} 2_{+} 1 V 2^{2} 2_{−} 1 V 22 22 22  22 22 22
V_{4} 7  196  22 22 22 22  22 22 22
2. 0_{+} 2_{+} 2_{+} -2 -19 21 1^{3} 2_{+} 1  1^{3} 2_{+} 1  3 22 22 3 22 3 3 3
A_{4} 2  18  3 22 22 3 3 22 3 3
3. 2_{−} 2_{+} 2_{−} -1 16 43 1^{3} 2_{+} 1  2^{2} 2_{−} 1 V 3 3 3 22  3 3 3
A_{4} 10  36  3 22 3 3 3 3 3 3

Discriminant class: 37

∞ 19 37 a_{1} a_{2} a_{3} Over p = 19  Over p = 37 2 3 5 7 11 13 17 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_{+} 2_{−} 3_{−}  -1 -180 178 2^{2} 2_{−} ∗ ∗ 1^{4} 3_{−} 37 37 4  4  22 4 4 4
C_{4}  7079  4 4  4  22 4 4