New Sextic Fields Unramified Outside of {∞, 2, 17}

Discriminant class: −34

∞  2 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 17  3 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_{i} 3_{−} -2 1 -8 1^{6} 11_{i} −2  1^{3} 2_{+} ∗  32 42 42 32 6 22 42 42
C_{3}^{2}.D_{4}  21 -14 3  1^{2} 1_{−} 17∗ 17∗ 42 42 32 3 222 222 33 42
2. 1_{k}  5_{k} 5_{−} -1 -1 -13 1^{3} 2_{+} ∗  1^{6} 5_{−} 17  6 42 42 6 32 22 42 42
C_{3}^{2}.D_{4}  8 4 -2 1^{2} 3_{k} −2∗ −2∗  42 42 6 33 2 2 3 42

Discriminant class: −2

∞  2 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 17 3 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}  11_{k} 2_{+} 0 -1 -4 1^{4} 9_{−} 2 −1 1^{3} 2_{+} ∗  42 32 6 42 32 42 222 32
C_{3}^{2}.D_{4}  12 -8  2 1^{2} 2_{i} −1 −1 2 0_{+} ∗ ∗ 6 2 33 42 222 32 42 32
4. 1_{k}  13_{i} 4_{+} -2 1 -20 1^{4} 10_{i} −1 2 1^{3} 2_{+} ∗  42 6 32 42 6 42 2 6
C_{3}^{2}.D_{4}  -19 162 -127 1^{2} 3_{+} 2  2 1^{3} 2_{+} ∗  32 222 3 42 2 6 42 6

Discriminant class: −1

∞  2 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 17 3 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. 1_{k}  14_{i} 2_{+} 2 -3 -12 1^{4} 11_{i} −2 2 1^{3} 2_{+} ∗  6 42 32 222 42 222 32 42
C_{3}^{2}.D_{4}  1 14 1 1^{2} 3_{+} 2  2 2 0_{+} ∗ ∗ 32 42 22 6 32 42 6 42
6. 3_{i}  14_{k} 4_{+} -2 -9 16 1^{4} 11_{−} 2 −2 1^{3} 2_{+} ∗  32 42 6 2 42 2 6 42
C_{3}^{2}.D_{4}  82 -4 114 1^{2} 3_{i} −2 −2 1^{3} 2_{+} ∗  6 42 22 32 6 42 32 42

Discriminant class: 2

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

Discriminant class: 17

∞  2 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 17  3 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
9. 2_{−} 14_{+} 3_{−} -2 -9 12 1^{4} 11_{+} 2  2 1^{3} 2_{+} ∗  6 6 2 222 42 42 32 222
C_{3}^{2}.D_{4}  21 -30 -49 1^{2} 3_{+} 2  2 1^{2} 1_{−} 17∗ 17∗ 2 6 32 42 33 42 42 222
10. 2_{−} 14_{+} 5_{−} -2 -21 32 1^{4} 11_{+} 2  2 1^{6} 5_{−} 17  32 32 222 2 42 42 6 2
C_{3}^{2}.D_{4}  2  94 -212 110 1^{2} 3_{+} 2  2  222 32 6 42 3 42 42 2

Discriminant class: 17 (continued)

∞  2 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 17  3 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
11. 2_{−} 12_{+} 3_{−} 0  2 -4 1^{4} 10_{k} −∗ −2 1^{3} 2_{+} ∗  6 6 6 32 5 5 4 4
S_{6}  1  4 -8 1^{2} 2_{i} −∗ −∗ 1^{2} 1_{−} 17∗ 17∗ 4 6 6 5 42 5 42 32
12. 2_{−} 12_{+} 5_{−} -2 13 -36 1^{4} 9_{k} −2 −∗ 1^{6} 5_{−} 17  32 32 32 6 5 5 4 4
S_{6}  2  60 -76 42 1^{2} 3_{i} −2 −2  4 32 32 5 42 5 42 6

Discriminant class: 34

∞  2 17 a_{1} a_{2} a_{3} Over p = 2  Over p = 17  3 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
13. 2_{−} 9_{+} 3_{−} -2 1 -2 3^{2} 9_{+} 2  1^{3} 2_{+} ∗  42 42 32 42 222 6 2 42
C_{3}^{2}.D_{4}  0 -4 -1  1^{2} 1_{−} 17∗ 17∗ 32 42 32 222 3 6 6 42
14. 2_{−} 3_{+} 5_{−} -1 -1 4 3 0_{+} 1  1^{6} 5_{−} 17  42 42 6 42 2 32 222 42
C_{3}^{2}.D_{4}  -9 4 -2 1^{2} 3_{+} 2 2  6 42 6 2 33 32 32 42