Old Sextic Fields Ramifying Above {2, 7}

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
1.  3_{i}  9_{+} 5_{k} 0 14 0 3^{2} 9_{+} 2  1^{6} 5_{k} 7∗  33 33 6  6 33 33 222
C_{6}  4  56 0 56  6 6 222 222 6 6 33 33
2.  1_{k}  9_{−} 5_{k} 0 0 0 3^{2} 9_{−} 2  1^{6} 5_{k} 7∗  33 33 6 22 6 33 33 2
A_{4} × C_{2}  -28 0 56  6 6 2 2 6 6 33 33

Discriminant class: −7

∞  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.  1_{k}  6_{−} 5_{k} 0 0 0 3^{2} 6_{−} 1  1^{6} 5_{k} 7∗  6 6 33 2 6 6 33 22
A_{4} × C_{2}  -7 0 7  6 33 2 22 6 33 6 6

Discriminant class: −2

∞  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
4.  3_{i}  9_{k} 4_{+} 0 10 0 3^{2} 9_{k} −2  2^{3} 4_{+} ∗  33 6 33 222 33 33 6 222
C_{6}  24 0 8  6 6  6 6 33 6
5.  1_{k}  9_{i} 4_{+} 0 -2 0 3^{2} 9_{i} −2  2^{3} 4_{+} ∗  33 6 33 2 33 33 6 2
A_{4} × C_{2}  -8 0 8  6 6 22 22 6 6 33 6

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
6.  3_{i}  6_{k} 4_{+} 0 5 0 3^{2} 6_{k} −1  2^{3} 4_{+} ∗  6 33 6  33 6 6
C_{6}  6 0 1  6 33  222 6 33 6 33
7.  1_{k}  6_{i} 4_{+} 0 -1 0 3^{2} 6_{i} −1  2^{3} 4_{+} ∗  6 33 6 22 33 6 6 22
A_{4} × C_{2}  -2 0 1  6 33 22 2 6 33 6 33
8.  3_{i}  12_{k} 4_{+} -2 3 -4 1^{4} 9_{+} 2∗ −1 1^{5} 4_{+} ∗  6 33 6 5 5 4 4 5
S_{5}  5 -6 7 1^{2} 3_{k} −2∗ −2∗  4 33 5 4 6 33 4 22

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
9. 2_{−} 6_{−} 4_{+} 0 1 0 3^{2} 6_{−} 1  1^{3} 2_{+} 1  33 33 33 22 33 33 33 22
A_{4}  -2 0 -1  1^{3} 2_{+} 1  33 33 22 22 33 33 33 33

Discriminant class: 2

∞  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
10.  0_{+}  9_{+} 4_{+} 0 -10 0 3^{2} 9_{+} 2  1^{3} 2_{+} 1  6 6 6 222 33 6 33 222
C_{6}  24 0 -8  1^{3} 2_{+} 1  33 6  222 33 6 6 6
11.  2_{−}  9_{−} 4_{+} 0 2 0 3^{2} 9_{−} 2  1^{3} 2_{+} 1  6 6 6 2 33 6 33 2
A_{4} × C_{2}  -8 0 -8  1^{3} 2_{+} 1  33 6 22 2 33 6 6 6

Discriminant class: 7

∞  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
12.  0_{+}  6_{k} 5_{i} 0 -7 0 3^{2} 6_{k} −1  1^{6} 5_{i} 7  33 6 6 222 6 33 6
C_{6}  14 0 -7  33 33 222 222 33 33 33 6
13.  2_{−}  6_{i} 5_{i} 0 0 0 3^{2} 6_{i} −1  1^{6} 5_{i} 7  33 6 6 2 6 33 6 22
A_{4} × C_{2}  -7 0 -7  33 33 2 2 33 33 33 6

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
14.  0_{+}  9_{k} 5_{i}  0 -14 0 3^{2} 9_{k} −2  1^{6} 5_{i} 7  6 33 33  6 6 6 222
C_{6}  56 0 -56  33 6 222  33 6 6 33
15.  2_{−}  9_{i} 5_{i}  0 0 0 3^{2} 9_{i} −2  1^{6} 5_{i} 7  6 33 33 22 6 6 6 2
A_{4} × C_{2}  -28 0 -56  33 6 2 22 33 6 6 33
16.  2_{−}  11_{i} 5_{i} -1 1 -8 1^{4} 11_{i} −2∗ −2 1^{6} 5_{i} 7  6 5 5 33 4 4 6 4
S_{5}  3  8 -8 8 2 0_{+} ∗  ∗  5 4 4 5 33 4 222 33