Quartic Fields Unramified Outside of {∞, 11, 37}

Discriminant class: −407

∞ 11 37 a_{1} a_{2} a_{3}  Over p = 11  Over p = 37 2 3 5 7 13 17 19 23
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d s 29 31 41 43 47 53 59 61
1. 1_{k} 1_{k} 3_{−} -1 5 -44 1^{2} 1_{k} 11 11 1^{4} 3_{−} 37 37 22  4 2 22 22 22 4
D_{4} 12 2 0_{+} ∗ ∗ 22 4 2 22 22 4 22

Discriminant class: −11

∞ 11 37 a_{1} a_{2} a_{3}  Over p = 11  Over p = 37 2 3 5 7 13 17 19 23
G h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d s 29 31 41 43 47 53 59 61
2. 1_{k} 1_{i} 2_{+} 2 -2 -3 1^{2} 1_{i} 11∗ 11∗ 1^{2} 1_{+} 37 37 4 22 22 2 4 4 4 22
D_{4} -7 1^{2} 1_{+} 37 37 4 22 2 4 22 22 4

Discriminant class: 1

∞ 11 37 a_{1} a_{2} a_{3}  Over p = 11  Over p = 37 2 3 5 7 13 17 19 23
G h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d s 29 31 41 43 47 53 59 61
3. 2_{−} 2_{−} 2_{+} 0 -13 0 1^{2} 1_{i} 11∗ 11∗ 1^{2} 1_{+} 37 37 22  22 22 22 22 22 22
V_{4} 8  144  1^{2} 1_{i} 11∗ 11∗ 1^{2} 1_{+} 37 37 22 22 22 22  22 22
4. 0_{+} 2_{+} 2_{+} -2 -9 -1 2^{2} 2_{+} 1  V 1^{3} 2_{+} 1  3 3 3 3 3 3 3 22
A_{4}  3  22 22 3 22  3 3 3

Discriminant class: 37

∞ 11 37 a_{1} a_{2} a_{3}  Over p = 11  Over p = 37 2 3 5 7 13 17 19 23
G h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d s 29 31 41 43 47 53 59 61
5. 2_{−} 0_{+} 3_{−}  -1  5 -7 2 0_{+} ∗  ∗ 1^{4} 3_{−} 37 37 4 22 4  4 4 4 4
C_{4}  49  2 0_{+} ∗  ∗  4 4 22 4  4 4
6. 0_{+} 2_{−} 3_{−}  1 -106 -104 1^{2} 1_{k} 11 11 1^{4} 3_{−} 37 37 4 22 4 22 4 4 4 4
C_{4}  2269  1^{2} 1_{k} 11 11  4 4  4  4 4
7. 2_{−} 2_{−} 1_{+}  -1  0 -4 1^{2} 1_{i} 11∗ 11∗ 1^{2} 1_{+} 37 37 4 22 2 22 4 4 4 2
D_{4}  5  1^{2} 1_{i} 11∗ 11∗  4 2 22 4  22 2 4
8. 2_{−} 2_{+} 3_{−}  2 20 19 2^{2} 2_{+} 1  V 1^{4} 3_{−} 37 37 2  4 22 2 2 2 4
D_{4} 16  192  2 4 22 2  22 4 2