Quartic Fields Ramifying Above {13, 29}

Discriminant class: 1

∞ 13 29 a_{1} a_{2} a_{3}  Over p = 13  Over p = 29 2 3 5 7 11 17 19 23
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
1. 0_{+} 2_{+} 2_{+} 0 -21 0 1^{2} 1_{+} 13 13 1^{2} 1_{+} 29 29 22 22 22 22 22 22 22
V_{4} 16 1^{2} 1_{+} 13 13 1^{2} 1_{+} 29 29 22 22 22 22 22 22 22

Discriminant class: 13

∞ 13 29 a_{1} a_{2} a_{3}  Over p = 13  Over p = 29 2 3 5 7 11 17 19 23
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
2. 2_{−} 3_{−} 2_{+} 1 93 -95 1^{4} 3_{−} 13 13 1^{2} 1_{+} 29 29 4 22 4 4 4  4 22
C_{4} 8 731 1^{2} 1_{+} 29 29 4 4 4 4 4 22
3. 2_{−} 1_{+} 2_{−} -1 -1 -3 1^{2} 1_{+} 13 13 2^{2} 2_{−} 1 V 4 22 2 2 4 22 4
D_{4} 9 4 4 4 22 4 22 2 22
4. 0_{+} 3_{−} 2_{−} 2 -18 -19 1^{4} 3_{−} 13 13 2^{2} 2_{−} 1 V 2 22 4 4 2 22 2 22
D_{4} 2 -4 2 2 2 22 2 22 4 22

Discriminant class: 29

∞ 13 29 a_{1} a_{2} a_{3}  Over p = 13  Over p = 29  2 3 5 7 11 17 19 23
G h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d  s 31 37 41 43 47 53 59 61
5. 2_{−} 2_{+} 3_{−} -1 91 -281 1^{2} 1_{−} 13∗ 13∗ 1^{4} 3_{−} 29 29 4 4  22 4 4 4
C_{4} 20 545 1^{2} 1_{−} 13∗ 13∗ 4 4 4 4 4 22 4
6. 2_{−} 2_{+} 1_{−} -2 6 -5 1^{2} 1_{+} 13 13 1^{2} 1_{−} 29∗ 29∗ 4 2 22 22 4 2 4
D_{4}  3  1^{2} 1_{+} 13 13 2 0_{+} ∗  ∗ 4 4 4 2 4 22 22 2
7. 0_{+} 2_{−} 3_{−} 0 -29  0 2^{2} 2_{−} 1  V 1^{4} 3_{−} 29 29 2 4 22 22 2 4 2
D_{4} 2  116  2 2 2 4 2 22 22 4

Discriminant class: 377

∞ 13 29  a_{1} a_{2} a_{3}  Over p = 13  Over p = 29  2 3 5 7 11 17 19 23
G h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d  s 31 37 41 43 47 53 59 61
8. 0_{+} 3_{−} 3_{−}  1 -141 -966 1^{4} 3_{−} 13 13 1^{4} 3_{−} 29 29 22 4 4 4  4 22 22
C_{4} 4  -1804  22 22  4 22  4 4
9. 0_{+} 3_{−} 3_{−}  -1 -141 589 1^{4} 3_{−} 13 13 1^{4} 3_{−} 29 29  4 4 4 22 4  22
C_{4} 4  -296  22 4  4 4
10. 0_{+} 3_{−} 1_{−}  1 -11 -4 1^{4} 3_{−} 13 13 1^{2} 1_{−} 29∗ 29∗ 22 2 4 4 22 2 22 22
D_{4} 16 2 0_{+} ∗ ∗ 22 22 22 2 22 22 4 2
11. 0_{+} 1_{−} 3_{−}  -2 -13 14 1^{2} 1_{−} 13∗ 13∗ 1^{4} 3_{−} 29 29 22 4 2 2 22 4 22
D_{4}  20  2 0_{+} ∗  ∗  22 22 22 4 22 22 2 4