Quartic Fields Unramified Outside of {∞, 17, 41}

Discriminant class: 1

∞ 17 41 a_{1} a_{2} a_{3} Over p = 17  Over p = 41 2 3 5 7 11 13 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. 0_{+} 2_{−} 2_{−} 0 -29 0 2^{2} 2_{−} 1 V 2^{2} 2_{−} 1 V  22 22 22 22 22 22 22
V_{4} 3 36 22 22 22 22 22 22

Discriminant class: 17

∞ 17 41 a_{1} a_{2} a_{3}  Over p = 17  Over p = 41 2 3 5 7 11 13 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
2. 0_{+} 3_{+} 0_{+}  1 -6 -1 1^{4} 3_{+} 17 17 4 0_{+} ∗ ∗ 22 4 4 4 4  22 4
C_{4}  1  4 4 4 22  22 22 4
3. 0_{+} 3_{+} 2_{+}  -1 -176 1 1^{4} 3_{+} 17 17 2^{2} 2_{+} ∗ ∗ 22 4 4 4 4 22  4
C_{4} 2  6971  4 4 4 22 22  22 4

Discriminant class: 41

∞ 17 41 a_{1} a_{2} a_{3} Over p = 17  Over p = 41 2 3 5 7 11 13 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
4. 0_{+} 0_{+} 3_{+}  -1 -15 -18 4 0_{+} ∗ ∗ 1^{4} 3_{+} 41 41 22 4 22 4 4 4 4
C_{4}  -4  4  22 4 4  22
5. 0_{+} 2_{+} 3_{+}  -1 -179 -346 2^{2} 2_{+} ∗ ∗ 1^{4} 3_{+} 41 41 22 4  4 4 4 4 22
C_{4} 2  2620  4 22 22 22 4 4
6. 0_{+} 2_{+} 3_{+}  0 -41 -123 1^{3} 2_{+} ∗  1^{4} 3_{+} 41 41 3 4 3 2 4 4 2 22
S_{4} 8  -82  2 3 3 22 2 4 3 3

Discriminant class: 697

∞ 17 41  a_{1} a_{2}  a_{3}  Over p = 17  Over p = 41  2 3 5 7 11 13 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
7. 0_{+} 3_{−} 3_{−}  -1 -261 1089 1^{4} 3_{−} 17∗ 17∗ 1^{4} 3_{−} 41∗ 41∗  4 22 22 4 4 4
C_{4} 12  324  4 4  4 4 22 4
8. 0_{+} 3_{−} 3_{−}  -1 -261 -1699 1^{4} 3_{−} 17∗ 17∗ 1^{4} 3_{−} 41∗ 41∗  22 4  4 4 4
C_{4} 12  -2464  22 4 4  4 4 22 4
9. 2_{−} 1_{+} 1_{−}  -1  2  -1 1^{2} 1_{+} 17 17 1^{2} 1_{−} 41∗ 41∗ 3 3 4 3 22 4 4 4
S_{4}  2  2 0_{+} ∗  ∗  3 2 2 3 4 2 3 4
10. 2_{−} 3_{−} 1_{+}  -1 -6  1 1^{4} 3_{−} 17∗ 17∗ 1^{2} 1_{+} 41 41 3 3 4 3 22 2 4 2
S_{4} 4  18  2 0_{+} ∗  ∗ 3 4 2 3 4 4 3 2
11. 2_{−} 3_{−} 1_{+}  -1 11  18 1^{4} 3_{−} 17∗ 17∗ 1^{2} 1_{+} 41 41 3 3 2 3 22 4 2 4
S_{4} 4  35  2 0_{+} ∗  ∗ 3 4 2 3 2 4 3 4