Quartic Fields Unramified Outside of {∞, 37, 41}

Discriminant class: 1

∞ 37 41 a_{1} a_{2} a_{3}  Over p = 37  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 29 31 43 47 53 59 61
1. 0_{+} 2_{+} 2_{+} 0 -39 0 1^{2} 1_{+} 37 37 1^{2} 1_{+} 41 41 22 22 22 22 22 22 22 22
V_{4} 1 1^{2} 1_{+} 37 37 1^{2} 1_{+} 41 41 22 22 22 22 22 22 22 22

Discriminant class: 37

∞ 37 41  a_{1} a_{2} a_{3}  Over p = 37  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 29 31 43 47 53 59 61
2. 2_{−} 3_{−} 0_{+}  -1 5 -7 1^{4} 3_{−} 37 37 2 0_{+} ∗  ∗ 4 22 4  22 4 4 4
C_{4}  49  2 0_{+} ∗  ∗ 4 4 4 4  4 4
3. 2_{−} 3_{−} 2_{+}  -1 375 -377 1^{4} 3_{−} 37 37 1^{2} 1_{−} 41∗ 41∗ 4  4 22  4 4 4
C_{4} 116  35939  1^{2} 1_{−} 41∗ 41∗ 4 4 4 4 22 22 4 4
4. 2_{−} 1_{−} 2_{+}  1 3 -14 1^{2} 1_{−} 37∗ 37∗ 1^{2} 1_{+} 41 41 2 22 2 22 22 4 4 4
D_{4} 3  32  2 0_{+} ∗  ∗ 1^{2} 1_{+} 41 41 2 4 2 2 22 22 2 2
5. 0_{+} 3_{−} 2_{−}  2 -54 -55 1^{4} 3_{−} 37 37 2^{2} 2_{−} 1  V 4 22 4 22 22 2 2 2
D_{4} 2  377  4 2 4 4 22 22 4 4

Discriminant class: 41

∞ 37 41  a_{1} a_{2} a_{3}  Over p = 37  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 29 31 43 47 53 59 61
6. 0_{+} 0_{+} 3_{+}  -1 -15 -18  1^{4} 3_{+} 41 41 22 4 22 4 4 4 4 4
C_{4}  -4  4  22 4 4  22
7. 0_{+} 2_{+} 3_{+}  1 -384 756 1^{2} 1_{+} 37 37 1^{4} 3_{+} 41 41  4  4 4 4 4 4
C_{4} 2  13280  1^{2} 1_{+} 37 37  22 4 22  4 4 22
8. 2_{−} 2_{−} 1_{+}  -1  4 -25 2^{2} 2_{−} 1 V 1^{2} 1_{+} 41 41 22 2 22 2 2 4 4 4
D_{4} 3  33  22 4 22 22 2 2 22 22
9. 2_{−} 2_{−} 3_{+}  0 41 0 2^{2} 2_{−} 1 V 1^{4} 3_{+} 41 41 22 4 22 4 4 2 2 2
D_{4} 32 41 22 2 22 22 4 4 22 22

Discriminant class: 1517

∞ 37 41  a_{1} a_{2}  a_{3}  Over p = 37  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 29 31 43 47 53 59 61
10. 2_{−} 3_{−} 3_{+}  1 190 -3508 1^{4} 3_{−} 37 37 1^{4} 3_{+} 41 41 4 4 4 4 4 22
C_{4} 136  71459  4  4 4 4 4 4 4
11. 2_{−} 3_{−} 3_{+}  -1 190 -5594 1^{4} 3_{−} 37 37 1^{4} 3_{+} 41 41 4 4 4 4 4  22 22
C_{4} 104  62357  4 22 4 4 4 4 4 4
12. 0_{+} 3_{−} 1_{−}  1 -32  44 1^{4} 3_{−} 37 37 1^{2} 1_{−} 41∗ 41∗ 4 2 4 2 2 22 22 22
D_{4}  49  2 0_{+} ∗  ∗ 4 22 4 4 2 2 4 4
13. 2_{−} 1_{−} 3_{+}  -2 22 -21 1^{2} 1_{−} 37∗ 37∗ 1^{4} 3_{+} 41 41 2 4 2 4 4 22 22 22
D_{4} 16  100  2 0_{+} ∗  ∗  2 22 2 2 4 4 2 2