Quartic Fields Ramifying Above {7, 41}

Discriminant class: 1

∞ 7 41 a_{1} a_{2} a_{3} Over p = 7  Over p = 41 2 3 5 11 13 17 19 23
G h a_{4} f^{e} c_{w} d s f^{e} c_{w} d s 29 31 37 43 47 53 59 61
1. 2_{−} 2_{+} 2_{−} 0 -17 0 2^{2} 2_{+} 1 V 2^{2} 2_{−} 1 V  22 22 22 22 22 22
V_{4} 7 144 22 22 22 22 22 22
2. 2_{−} 2_{+} 2_{−} -1 4 -7 1^{3} 2_{+} 1  2^{2} 2_{−} 1 V 3 3 3 3 22 3 3 3
A_{4} 2 8 3 3 22 3 3 3 3

Discriminant class: 41

∞ 7 41 a_{1} a_{2} a_{3} Over p = 7  Over p = 41 2 3 5 11 13 17 19 23
G h a_{4} f^{e} c_{w} d s f^{e} c_{w} d s 29 31 37 43 47 53 59 61
3. 2_{−} 2_{−} 3_{+} -1 67 146 2^{2} 2_{−} ∗ ∗ 1^{4} 3_{+} 41 41 22 4  4 4 4 4
C_{4} 10  652  4 22  22 4 4 22