Old Sextic Fields Ramifying Above {7, 17}

Discriminant class: −119

∞ 7 17 a_{1} a_{2} a_{3} Over p = 7  Over p = 17 2 3 5 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. 3_{i} 5_{i} 3_{−} -1 29 -29 1^{6} 5_{i} 7  3^{2} 3_{−} 17∗  33 33 33 6 222 6 6 222
C_{6} 10  253 -253 701  33 6  6 33 6 33
2. 3_{i} 3_{k} 5_{+} -3 25 -45 2^{2} 2_{+} 1 V 1^{6} 5_{+} 17∗  33 22 22 222 222 222 6 6
D_{6} 10 155 -133 123 1^{2} 1_{k} 7 7 22 6 22 33 222 33 222 22

Discriminant class: −7

∞  7 17 a_{1} a_{2} a_{3}  Over p = 7  Over p = 17 2 3 5 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
3.  3_{i}  3_{k} 4_{+} 3 -7 -19 1^{2} 1_{i} 7∗ 7∗ 2^{3} 4_{+} ∗  33 222 222  222 222 33 33
S_{3}  -5 5 121 1^{2} 1_{i} 7∗ 7∗  222 33 222 33 222 33 222 222
4.  3_{i}  5_{k} 4_{+} -1 1 48 1^{6} 5_{k} 7∗  2^{3} 4_{+} ∗  3 6 6 33 222 6 3 33
S_{3} × C_{3} 3  225 188 64  6 3 222 33 6 3 6 6

Discriminant class: 17

∞ 7 17 a_{1} a_{2} a_{3} Over p = 7  Over p = 17 2 3 5 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
5. 0_{+} 4_{+} 3_{+} -1 -17 11 2^{3} 4_{+} ∗  3^{2} 3_{+} 17  33 6 6 6  33 6 222
C_{6}  57 -25 -13  6 6 222  33 33 33 6
6. 2_{−} 2_{+} 5_{−} -1 -1 21 2^{2} 2_{+} 1 V 1^{6} 5_{−} 17  33 222 222 222 22 22 6 6
D_{6}  25 -13 -19 2 0_{+} ∗ ∗  222 6 222 33 22 33 22 222