Quartic Fields Ramifying Above {31, 37}

Discriminant class: −1147

∞ 31 37  a_{1} a_{2} a_{3}  Over p = 31  Over p = 37  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 41 43 47 53 59 61
1. 1_{k} 3_{i} 1_{+}  -1 -19 -33 1^{4} 3_{i} 31 31∗ 1^{2} 1_{+} 37 37 4 2 4 3 2 3 22 2
S_{4}  -27  2 0_{+} ∗  ∗ 3 3 3 3 3 2 2 3
2. 1_{k} 1_{i} 3_{+}  2 20 93 1^{2} 1_{i} 31∗ 31∗ 1^{4} 3_{+} 37∗ 37∗ 4 2 2 3 4 3  4
S_{4} 2  81  2 0_{+} ∗  ∗  3 3 3 3 3 4 2 3
3. 1_{k} 3_{i} 3_{+}  -1 -143 1362 1^{4} 3_{i} 31 31∗ 1^{4} 3_{+} 37∗ 37∗ 2 2 4 3 4 3 22 4
S_{4} 4 -3096 3 3 3 3 3 4 2 3

Discriminant class: −31

∞ 31 37 a_{1} a_{2} a_{3}  Over p = 31  Over p = 37  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 41 43 47 53 59 61
4. 1_{k} 1_{k} 2_{−} -1 3 -6 1^{2} 1_{k} 31 31 1^{2} 1_{−} 37∗ 37∗ 3 4 3 3 4 4 2 3
S_{4} 2 -1 2 0_{+} ∗ ∗ 1^{2} 1_{+} 37 37 2 4 3 2 2 3 2

Discriminant class: 1

∞ 31 37 a_{1} a_{2} a_{3}  Over p = 31  Over p = 37 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 41 43 47 53 59 61
5. 2_{−} 2_{+} 2_{−} 0 -3 0 2^{2} 2_{+} 1 V 2^{2} 2_{−} 1 V 22 22 22  22 22 22 22
V_{4} 9 289 22 22 22 22 22 22
6. 2_{−} 2_{−} 2_{+} -1 10 -1 1^{2} 1_{k} 31 31 1^{3} 2_{+} 1  3 3 3 3 22 3 3 3
A_{4} 4  32  1^{2} 1_{k} 31 31  22 22 3  22 3 3 3
7. 0_{+} 2_{+} 2_{+} 2 -25 3 1^{3} 2_{+} 1  1^{3} 2_{+} 1  3 22 3 22 3 3 3 22
A_{4}  45  22 22 3 3 22 22 3 3

Discriminant class: 37

∞ 31 37  a_{1} a_{2} a_{3} Over p = 31  Over p = 37 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 41 43 47 53 59 61
8. 0_{+} 2_{−} 3_{−}  1 -291 -289 2^{2} 2_{−} ∗ ∗ 1^{4} 3_{−} 37 37 4  4  4 4 4
C_{4}  19289  4 4 22 4  22 4 4