Old Sextic Fields Ramifying Above {5, 17}

Discriminant class: 5

∞ 5 17 a_{1} a_{2} a_{3} Over p = 5  Over p = 17 2 3 7 11 13 19 23 29
G h  a_{4} a_{5} a_{6} f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
1. 2_{−} 5_{−} 4_{+} -1  -5  0 1^{5} 5_{−} 5  1^{5} 4_{+} ∗  6 222 6 22 6 5 4 5
S_{5}  -5 -16  1  33 4 33 4 222 4 33 22
2. 2_{−} 9_{−} 2_{+} -1  5 15 1^{5} 9_{−} 5  2^{2} 2_{+} ∗ ∗ 4 4 222 33 222 33 6 5
S_{5}  25 -13 -12  5 222 5 4 4 4 5 5
3. 2_{−} 7_{−} 4_{+} 3  0 -10 1^{5} 7_{−} 5  1^{5} 4_{+} ∗  6 4 6 33 4 5 222 22
S_{5}  -15 59 117  5 6 5 4 6 4 33 22
4. 2_{−} 9_{−} 4_{+} -2  -5 20 1^{5} 9_{−} 5  1^{5} 4_{+} ∗  6 6 4 5 6 22 4 5
S_{5}  -25 -411 847  22 4 22 4 4 222 33 33
5. 2_{−} 9_{−} 4_{+} -1  0 -25 1^{5} 9_{−} 5  2^{3} 4_{+} ∗  222 6 4 5 6 33 4 33
S_{5}  0 -225 -25  5 6 33 6 4 222 5 5
6. 2_{−} 9_{−} 4_{+} 1 15 120 1^{5} 9_{−} 5  2^{3} 4_{+} ∗  6 4 222 5 6 5 4 5
S_{5}  215 1386 1251  5 4 33 4 222 4 33 5

Discriminant class: 17

∞ 5 17 a_{1} a_{2}  a_{3} Over p = 5  Over p = 17 2 3 7 11 13 19 23 29
G h  a_{4} a_{5}  a_{6} f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
7. 2_{−} 6_{+} 3_{−} -2  5  10 1^{5} 6_{+} ∗  2^{2} 2_{−} 1 V 5 6 4 4 22 5 4 4
S_{5} 2  -10 11  -2  1^{2} 1_{+} 17 17 6 6 6 33 33 33 33 222
8. 2_{−} 4_{+} 5_{−} -2 -4  32 2^{3} 4_{+} ∗  1^{6} 5_{−} 17  5 4 6 4 5 33 6 4
S_{5} 2  -59 43  -9  4 4 6 5 33 5 22 4
9. 2_{−} 6_{+} 5_{−} -1 -35 -115 1^{5} 6_{+} ∗  1^{6} 5_{−} 17  22 6 6 4 5 5 4 6
S_{5} 2  -145 -47  32  4 222 4 33 5 33 33 222
10. 2_{−} 8_{+} 5_{−} -1 -35  55 1^{5} 8_{+} ∗  1^{6} 5_{−} 17  33 4 4 222 5 33 4 222
S_{5} 2  365 89 -699  6 4 4 22 22 5 5 6
11. 2_{−} 8_{+} 5_{−} -1 -35  55 1^{5} 8_{+} ∗  1^{6} 5_{−} 17  22 4 6 4 22 5 222 4
S_{5} 2  365 -421 -1464  4 6 222 5 33 33 22 4

Discriminant class: 85

∞ 5 17 a_{1} a_{2} a_{3}  Over p = 5  Over p = 17  2 3 7 11 13 19 23 29
G h  a_{4} a_{5} a_{6} f^{e} c_{w} d s f^{e} c_{w} d  s 31 37 41 43 47 53 59 61
12. 2_{−} 7_{+} 3_{−} 0 5 0 1^{5} 7_{+} 5∗  1^{4} 3_{−} 17∗ 17∗ 6 5 5 6 222 5 5 4
S_{5}  15 13 -5  4 5 6 6 4 6 5 4