Quartic Fields Unramified Outside of {∞, 5, 41}

Discriminant class: 1

∞ 5 41 a_{1} a_{2} a_{3} Over p = 5  Over p = 41 2 3 7 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. 0_{+} 2_{+} 2_{+} 0 -23 0 1^{2} 1_{+} 5 5 1^{2} 1_{+} 41 41 22 22 22 22 22 22 22 22
V_{4} 81 1^{2} 1_{+} 5 5 1^{2} 1_{+} 41 41 22 22 22 22 22

Discriminant class: 5

∞ 5 41 a_{1} a_{2} a_{3} Over p = 5  Over p = 41 2 3 7 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
2. 2_{−} 3_{−} 0_{+} -1 1 -1 1^{4} 3_{−} 5 5  4 4 4  4 4 22 4
C_{4}  1  22  4 4 4 4 22
3. 2_{−} 3_{−} 2_{+} -1 51 -51 1^{4} 3_{−} 5 5 1^{2} 1_{+} 41 41 4 4 4 22 4 4  4
C_{4} 8 551 1^{2} 1_{+} 41 41 4 4 4 4 22
4. 2_{−} 1_{+} 2_{−} -2 8 -7 1^{2} 1_{+} 5 5 2^{2} 2_{−} 1 V 2 4 4 22 4 4 22 2
D_{4} 2 22 22 2 2 4 4 22
5. 0_{+} 3_{−} 2_{−} 0 -15 0 1^{4} 3_{−} 5 5 2^{2} 2_{−} 1 V 4 2 2 22 2 2 22 4
D_{4} 2 5 22 22 4 4 2 2 22 22

Discriminant class: 41

∞ 5 41 a_{1} a_{2} a_{3}  Over p = 5  Over p = 41  2 3 7 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
6. 0_{+} 0_{+} 3_{+} -1 -15 -18 2 0_{+} ∗ ∗ 1^{4} 3_{+} 41 41 22 4 4 4 4 4 4
C_{4} -4 2 0_{+} ∗ ∗ 4 22 4 4 22
7. 0_{+} 2_{+} 3_{+} 1 -56 100 1^{2} 1_{−} 5∗ 5∗ 1^{4} 3_{+} 41 41  4 4 4 4 4 4 22
C_{4} 2 160 1^{2} 1_{−} 5∗ 5∗ 4 22 4 4 22
8. 2_{−} 2_{+} 1_{−} -1 3 -2 1^{2} 1_{+} 5 5 1^{2} 1_{−} 41∗ 41∗ 22 4 4 2 4 4 2 22
D_{4} 4 1^{2} 1_{+} 5 5 2 0_{+} ∗ ∗ 2 22 22 22 4 4 22
9. 2_{−} 2_{−} 3_{+} -1 -15 -18 2^{2} 2_{−} 1 V 1^{4} 3_{+} 41 41 22 2 2 4 2 2 4 22
D_{4} 12 324 4 22 22 22 2 2

Discriminant class: 205

∞ 5 41 a_{1} a_{2} a_{3}  Over p = 5  Over p = 41  2 3 7 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
10. 2_{−} 3_{−} 3_{+}  1 26 -64 1^{4} 3_{−} 5 5 1^{4} 3_{+} 41 41 4  22 4  4 4
C_{4} 40  1431  4  4 4  22 22
11. 2_{−} 3_{−} 3_{+}  -1 26 -346 1^{4} 3_{−} 5 5 1^{4} 3_{+} 41 41 4 22  4 22 22 4 4
C_{4} 8  1021  4  4 4 22 22 22 22
12. 0_{+} 3_{−} 1_{−}  -2 -6  7 1^{4} 3_{−} 5 5 1^{2} 1_{−} 41∗ 41∗ 4 22 22 2 22 22 2 4
D_{4}  11  2 0_{+} ∗  ∗ 2 22 4 4 22 22 22 22
13. 2_{−} 1_{−} 3_{+}  -2 22 -21 1^{2} 1_{−} 5∗ 5∗ 1^{4} 3_{+} 41 41 2 22 22 4 22 22 4 2
D_{4} 6  18  2 0_{+} ∗ ∗  4 22 2 2 22 22