Old Sextic Fields Ramifying Above {17, 19}

Discriminant class: −323

∞ 17 19 a_{1} a_{2} a_{3} Over p = 17  Over p = 19 2 3 5 7 11 13 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} 3_{+} 5_{k} -1 78 84 3^{2} 3_{+} 17  1^{6} 5_{k} 19∗  6 33 6 222 222 6 6 33
C_{6} 28  607 603 2287  33 33 33 6 6 6

Discriminant class: −19

∞ 17 19 a_{1} a_{2} a_{3} Over p = 17  Over p = 19 2 3 5 7 11 13 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
2. 3_{i} 4_{+} 5_{k} -1 21 46 1^{3} 2_{+} ∗  1^{6} 5_{k} 19∗  6 4 22 5 5 6 22 6
S_{5} 24  37 128 159 1^{3} 2_{+} ∗  222 4 4 5 33 4 4 5

Discriminant class: 17

∞ 17 19 a_{1} a_{2} a_{3} Over p = 17  Over p = 19 2 3 5 7 11 13 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. 0_{+} 3_{+} 4_{+} -1 -25 7 3^{2} 3_{+} 17  1^{3} 2_{+} 1  33 6 6 222 222 33 6 6
C_{6}  95 -95 19  1^{3} 2_{+} 1  222 222 6 33 33 33 33 6