Quartic Fields Unramified Outside of {∞, 13, 47}

Discriminant class: 1

∞ 13 47 a_{1} a_{2} a_{3} Over p = 13  Over p = 47 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 37 41 43 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} 25  225  22 22 22 22 22  22
2. 0_{+} 2_{+} 2_{+} -1 -17 38 1^{3} 2_{+} 1  2^{2} 2_{+} 1 V 3 3 22 3 3 3 3 3
A_{4} -13 3 22 3 3 3 22 3 3

Discriminant class: 13

∞ 13 47 a_{1} a_{2} a_{3}  Over p = 13  Over p = 47 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 37 41 43 53 59 61
3. 2_{−} 3_{−} 0_{+}  -1  2  4 1^{4} 3_{−} 13 13 4 0_{+} ∗ ∗ 4  4 4 4 22 4 22
C_{4}  3  4 4 4 22  4
4. 0_{+} 3_{−} 2_{−}  -1 -154 -152 1^{4} 3_{−} 13 13 2^{2} 2_{−} ∗ ∗ 4  4 4 4 22 4
C_{4}  1719  22 4 4 4  4