New Sextic Fields Ramifying Above {3, 17}

Discriminant class: −51

∞  3 17 a_{1} a_{2} a_{3}  Over p = 3  Over p = 17  2 5 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. 3_{i} 11_{k} 3_{+} -3 15 -16 1^{6} 11_{k} 3  1^{3} 2_{+} ∗  6 33 32 5 5 5 5 5
S_{6}  2  51 6 79  1^{2} 1_{+} 17 17 32 32 42 5 2 32 6 6
2. 1_{k} 9_{i} 5_{+} 3 -9 -6 1^{3} 5_{i} 3∗  1^{6} 5_{+} 17∗  32 3 6 5 5 5 5 5
S_{6}  2  87 48 -64 1^{3} 4_{+} ∗  6 6 42 5 222 6 32 32

Discriminant class: −3

∞ 3 17 a_{1} a_{2} a_{3} Over p = 3 Over p = 17 2 5 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
3. 3_{i} 5_{k} 4_{+} -1 -2 1 1^{3} 4_{+} ∗  1^{5} 4_{+} ∗  4 6 5 6 42 5 32 6
S_{6}  14 -17 6 1^{2} 1_{k} 3 3  5 42 32 42 6 32 6 5
4. 1_{k} 9_{i} 4_{+} 3 12 -19 1^{6} 9_{i} 3∗  1^{5} 4_{+} ∗  4 32 5 32 42 5 6 32
S_{6}  -42 39 -8  5 42 6 42 32 6 32 5

Discriminant class: 1

∞  3 17 a_{1} a_{2} a_{3} Over p = 3  Over p = 17 2 5 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
5. 2_{−} 6_{−} 4_{+} 0 -6 -4 2^{3} 6_{−} 1  1^{4} 3_{+} 17 17 33 42 42 42 3 22 42 42
C_{3}^{2}.C_{4} 9 12 -13 1^{2} 1_{+} 17 17 42 42 42 22 22 33 3 42
6. 2_{−} 6_{−} 4_{+} 0 6 -1 2^{3} 6_{−} 1  1^{4} 3_{+} 17 17 3 42 42 42 33 22 42 42
C_{3}^{2}.C_{4} 9 -3 -4 1^{2} 1_{+} 17 17 42 42 42 22 22 3 33 42

Discriminant class: 17

∞  3 17 a_{1} a_{2} a_{3}  Over p = 3  Over p = 17  2 5 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_{−} 10_{+} 3_{−} 0 0 -15 1^{6} 10_{+} ∗  1^{4} 3_{−} 17∗ 17∗ 5 32 4 32 5 33 6 6
S_{6} 27 -54 63 2 0_{+} ∗ ∗ 32 32 32 42 22 42 5 6
8. 2_{−} 10_{+} 3_{−} 0 -6 -11 1^{6} 10_{+} ∗  1^{4} 3_{−} 17∗ 17∗ 5 6 4 6 5 3 32 32
S_{6}  -45 -66 -29  2 0_{+} ∗  ∗ 6 6 6 42 22 42 5 32
9. 2_{−} 10_{+} 3_{−} 3 9 18 1^{6} 10_{+} ∗  1^{3} 2_{+} ∗  5 32 6 32 33 22 4 222
S_{6}  0 0 -51  1^{2} 1_{−} 17∗ 17∗ 32 32 4 22 5 42 42 4
10. 2_{−} 10_{+} 5_{−} 3 -9 -40 1^{6} 10_{+} ∗  1^{6} 5_{−} 17  5 6 32 6 3 22 4 2
S_{6}  2  -15 99 208  6 6 4 22 5 42 42 4