Quartic Fields Unramified Outside of {∞, 29, 41}

Discriminant class: 1

∞ 29 41 a_{1} a_{2} a_{3} Over p = 29  Over p = 41 2 3 5 7 11 13 17 19
G h a_{4} f^{e} c_{w} d s f^{e} c_{w} d s 23 31 37 43 47 53 59 61
1. 0_{+} 2_{−} 2_{−} 0 -35 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: 29

∞ 29 41 a_{1} a_{2} a_{3}  Over p = 29  Over p = 41 2 3 5 7 11 13 17 19
G h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d s 23 31 37 43 47 53 59 61
2. 2_{−} 3_{−} 0_{+}  -1 4 -20 1^{4} 3_{−} 29 29 4 0_{+} ∗ ∗ 4 4 22  4 22 4 4
C_{4}  23  4 4 4 4  4
3. 2_{−} 3_{−} 2_{+}  -1 294 -890 1^{4} 3_{−} 29 29 2^{2} 2_{+} ∗ ∗ 4 4 22 22 4  4 4
C_{4} 34  3793  4 4 4 4 22  4
4. 2_{−} 1_{−} 2_{+}  -1 4 -6 1^{2} 1_{−} 29∗ 29∗ 1^{3} 2_{+} ∗  4 4 3 3 4 3 4 2
S_{4}  7  2 0_{+} ∗  ∗  22 4 4 2 4 22 3 2

Discriminant class: 41

∞ 29 41 a_{1} a_{2} a_{3} Over p = 29  Over p = 41 2 3 5 7 11 13 17 19
G h a_{4} f^{e} c_{w} d s f^{e} c_{w} d s 23 31 37 43 47 53 59 61
5. 0_{+} 0_{+} 3_{+}  -1 -15 -18 4 0_{+} ∗ ∗ 1^{4} 3_{+} 41 41 22 4 22 4 4 4 4 4
C_{4}  -4  22 4 4  22
6. 0_{+} 2_{+} 3_{+}  1 -302 592 2^{2} 2_{+} ∗ ∗ 1^{4} 3_{+} 41 41  4 22 4 4 4 4 4
C_{4} 2  8032  22 22  4 4

Discriminant class: 1189

∞ 29 41  a_{1} a_{2} a_{3}  Over p = 29  Over p = 41  2 3 5 7 11 13 17 19
G  h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d  s 23 31 37 43 47 53 59 61
7. 2_{−} 3_{+} 3_{−}  1 149 2601 1^{4} 3_{+} 29∗ 29∗ 1^{4} 3_{−} 41∗ 41∗ 4 22  4 22 4
C_{4} 100  44973  4 4 4 22 4  4
8. 2_{−} 3_{+} 3_{−}  1 149 4979 1^{4} 3_{+} 29∗ 29∗ 1^{4} 3_{−} 41∗ 41∗ 4  4  4 22 22
C_{4} 388  30705  4 4 4  4  4