Quartic Fields Unramified Outside of {∞, 31, 41}

Discriminant class: −1271

∞ 31 41 a_{1} a_{2} a_{3}  Over p = 31  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 37 43 47 53 59 61
1. 1_{k} 1_{i} 3_{+} -2 63 -62 1^{2} 1_{i} 31∗ 31∗ 1^{4} 3_{+} 41 41  22 22 4 22 22 22 4
D_{4} 8 -64 2 22 2 2 4 22 22 2

Discriminant class: −31

∞ 31 41 a_{1} a_{2} a_{3}  Over p = 31  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 37 43 47 53 59 61
2. 1_{k} 1_{i} 2_{+} -1 1 -26 1^{2} 1_{i} 31∗ 31∗ 1^{2} 1_{+} 41 41 22 4  22 4 4 4 22
D_{4} 20 1^{2} 1_{+} 41 41 2 4 2 2 22 4 22 2

Discriminant class: 1

∞ 31 41 a_{1} a_{2} a_{3}  Over p = 31  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 37 43 47 53 59 61
3. 2_{−} 2_{−} 2_{+} 0 -5 0 1^{2} 1_{i} 31∗ 31∗ 1^{2} 1_{+} 41 41  22  22 22 22 22 22
V_{4} 60  324  1^{2} 1_{i} 31∗ 31∗ 1^{2} 1_{+} 41 41 22 22 22 22 22 22  22

Discriminant class: 41

∞ 31 41  a_{1} a_{2} a_{3}  Over p = 31  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 37 43 47 53 59 61
4. 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
5. 2_{−} 2_{−} 3_{+}  -1 313 638 1^{2} 1_{i} 31∗ 31∗ 1^{4} 3_{+} 41 41 22 4 22 4 4 4 4 4
C_{4} 16  10492  1^{2} 1_{i} 31∗ 31∗  22 4 22  4 4
6. 2_{−} 2_{−} 1_{+}  -1 -2  5 1^{2} 1_{i} 31∗ 31∗ 1^{2} 1_{+} 41 41 22 4  2 4 4 4 2
D_{4} 3  25  1^{2} 1_{i} 31∗ 31∗  22 4 22 22 2 4 22 22
7. 2_{−} 2_{−} 3_{+}  2 -101 -102 1^{2} 1_{i} 31∗ 31∗ 1^{4} 3_{+} 41 41  2 22 4 2 2 2 4
D_{4} 320  3872  1^{2} 1_{i} 31∗ 31∗  22 2 22 22 4 2 22 22