Quartic Fields Ramifying Above {17, 29}

Discriminant class: 1

∞ 17 29 a_{1} a_{2} a_{3} Over p = 17  Over p = 29 2 3 5 7 11 13 19 23
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 -23 0 2^{2} 2_{−} 1 V 2^{2} 2_{−} 1 V 22 22 22 22 22  22 22
V_{4} 9 22 22 22 22 22 22

Discriminant class: 17

∞ 17 29 a_{1} a_{2} a_{3}  Over p = 17  Over p = 29 2 3 5 7 11 13 19 23
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
2. 0_{+} 3_{+} 2_{+}  -1 -125 1 1^{4} 3_{+} 17 17 2^{2} 2_{+} ∗ ∗  4 4 4 4  4
C_{4} 2  3452  4 4 4  22 22 22 4

Discriminant class: 29

∞ 17 29 a_{1} a_{2} a_{3} Over p = 17  Over p = 29 2 3 5 7 11 13 19 23
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_{−} 2_{+} 3_{−} 1 120 368 2^{2} 2_{+} ∗ ∗ 1^{4} 3_{−} 29 29 4 4  22 4 22 4 22
C_{4} 18 835 4 4 4 4 4 4

Discriminant class: 493

∞ 17 29 a_{1} a_{2} a_{3}  Over p = 17  Over p = 29  2 3 5 7 11 13 19 23
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_{−} 3_{−} 3_{+}  -1 62 154 1^{4} 3_{−} 17∗ 17∗ 1^{4} 3_{+} 29∗ 29∗ 4  4 4 22 22 4 4
C_{4} 52  8433  22  4 4 22 22 22
5. 2_{−} 3_{−} 3_{+}  -1 62 -832 1^{4} 3_{−} 17∗ 17∗ 1^{4} 3_{+} 29∗ 29∗ 4 22 4 4  22 4 4
C_{4} 20  7447  22  22 4 4 22 22