New Sextic Fields Ramifying Above {2, 7}

Discriminant class: 1

∞  2 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 7 3 5 11 13 17 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_{−} 14_{−} 2_{+} -2 -1 0 1^{4} 11_{−} 2 2 1^{3} 2_{+} 1  42 42 42 42 33 42 33 42
C_{3}^{2}.C_{4} 6 -4 2 1^{2} 3_{+} 2 2 3 0_{+} 1 22 42 22 42 22 42 42 42
2. 2_{−} 14_{−} 4_{+} -2 1 -4 1^{4} 11_{−} 2 2 1^{3} 2_{+} 1  42 42 42 42 3 42 3 42
C_{3}^{2}.C_{4} 3 -11 10 41 1^{2} 3_{+} 2 2 1^{3} 2_{+} 1 22 42 22 42 22 42 42 42

Discriminant class: 14

∞  2 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 7 3 5 11 13 17 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. 2_{−} 13_{k} 3_{k} -2 3 4 1^{4} 10_{k} −1 2 1^{3} 2_{+} 1  6 42 42 42 32 222 32 6
C_{3}^{2}.D_{4} 2 4 2 1^{2} 3_{+} 2 2 1^{2} 1_{k} 7 7 33 6 32 42 3 6 6 42
4. 2_{−} 11_{i} 5_{i} -2 4 -8 1^{4} 9_{+} 2 −1 1^{6} 5_{i} 7  32 42 42 42 6 2 6 32
C_{3}^{2}.D_{4}  3  9 10 -6 1^{2} 2_{i} −1 −1  3 32 6 42 33 32 32 42