New Sextic Fields Ramifying Above {3, 19}

Discriminant class: −19

∞  3 19 a_{1} a_{2} a_{3}  Over p = 3  Over p = 19  2 5 7 11 13 17 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}  10_{+} 1_{k} 0 3 4 1^{6} 10_{+} ∗  1^{2} 1_{k} 19 19 6 42 22 42 32 42 42 222
C_{3}^{2}.D_{4}  9 -3 7  2 0_{+} ∗  ∗ 2 32 6 3 42 6 6 22
2. 1_{k}  10_{+} 3_{i} 0 -6 -7 1^{6} 10_{+} ∗  2^{2} 2_{+} 1  V 32 42 22 42 6 42 42 2
C_{3}^{2}.D_{4}  9 21 -2  1^{2} 1_{i} 19∗ 19∗ 222 6 32 33 42 32 32 22

Discriminant class: 1

∞  3 19 a_{1} a_{2} a_{3}  Over p = 3  Over p = 19 2 5 7 11 13 17 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. 2_{−} 6_{−} 4_{+} -3 -3 14 1^{3} 5_{i} 3∗  1^{5} 4_{+} 1  5 42 5 5 42 5 42 33
A_{6}  0 -12 9 1^{2} 1_{i} 3∗ 3∗  42 42 5 5 5 5 42 5
4. 2_{−} 10_{−} 4_{+} 0 -3 -3 1^{3} 5_{i} 3∗  1^{5} 4_{+} 1  5 42 5 5 42 5 42 3
A_{6}  3  21 27 -16 1^{3} 5_{i} 3∗  42 42 5 5 5 5 42 5