Old Sextic Fields Unramified Outside of {∞, 5, 7}

Discriminant class: −35

∞ 5 7 a_{1} a_{2} a_{3} Over p = 5 Over p = 7 2 3 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. 3_{i} 3_{−} 5_{i} 1 8 8 3^{2} 3_{−} 5∗  1^{6} 5_{i} 7  6 33 33  33 6 6
C_{6} 2  22 22 29  6 6 222 222 33 6 6 6
2. 3_{i} 5_{+} 3_{k} -1 5 0 1^{6} 5_{+} 5∗  2^{2} 2_{+} 1 V 6 22 33 22 22 222 6 33
D_{6} 2 5 4 1 1^{2} 1_{k} 7 7 222 6 222 6 22 222 222 222
3. 3_{i} 7_{+} 3_{k} 0 -5 0 1^{5} 7_{+} 5∗  1^{4} 3_{k} 7∗ 7 6 33 22 33 5 4 4 22
S_{5} 15 9 5 2 0_{+} ∗ ∗ 4 6 6 222 22 6 6 6

Discriminant class: −7

∞  5 7 a_{1} a_{2} a_{3} Over p = 5  Over p = 7  2 3 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
4.  3_{i}  0_{+} 5_{k} -1 1 -1 6 0_{+} ∗  1^{6} 5_{k} 7∗  33 6 33 222 6 6 33
C_{6}  1 -1 1  6 33 222  6 33 6 6
5.  3_{i}  4_{+} 3_{k} -3 1 3 2^{3} 4_{+} ∗  1^{2} 1_{i} 7∗ 7∗ 33 222 33 222 222 222 33 33
S_{3}  1 -3 1  1^{2} 1_{i} 7∗ 7∗ 222 33 222 33 222  222 222
6.  3_{i}  4_{+} 5_{k} 1 1 -6 2^{3} 4_{+} ∗  1^{6} 5_{k} 7∗  3 6 3 222 6 6 3 33
S_{3} × C_{3} 3  8 29 22  6 3 222 33 6 33 6 6

Discriminant class: 5

∞ 5 7 a_{1} a_{2} a_{3} Over p = 5  Over p = 7 2 3 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
7. 0_{+} 3_{+} 4_{+} 1 -7 -2 3^{2} 3_{+} 5  2^{3} 4_{+} ∗  6 6 33 222 6 33 6
C_{6}  7 2 -1  33 6  222 6 6 33 33
8. 2_{−} 5_{−} 2_{+} -1 0 0 1^{6} 5_{−} 5  2^{2} 2_{+} 1 V 6 222 33 222 222 22 6 33
D_{6} 0 4 1 2 0_{+} ∗ ∗ 22 6 22 6 222 222 22 22
9. 2_{−} 9_{−} 2_{+} -2 0 -10 1^{5} 9_{−} 5  2^{2} 2_{+} 1 V 4 6 33 6 4 5 4 5
S_{5}  10 1 -2  2 0_{+} ∗ ∗ 33 222 33 4 6 222 5 22
10. 2_{−} 7_{−} 4_{+} -2 0 -5 1^{5} 7_{−} 5  2^{3} 4_{+} ∗  6 4 22 6 4 5 6 33
S_{5} 3  0 20 -15  5 4 22 6 222 4 22 5
11. 2_{−} 9_{−} 4_{+} -2 0 -10 1^{5} 9_{−} 5  1^{5} 4_{+} ∗  222 4 5 222 4 33 4 33
S_{5}  10 -4 33  33 4 5 222 6 6 22 5