Septic Fields Unramified Outside of {∞, 2, 3}

Discriminant class: −6

∞  2  3 a_{1} a_{2} a_{3} a_{4}  Over p = 2  Over p = 3 5 7 11 13 17 19 23 29
G  h  a_{5} a_{6} a_{7}  f^{e} c_{w} d  s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
1. 3_{i} 9_{+} 11_{k} 1 3 5 2 3^{2} 9_{+} 2∗  1^{6} 11_{k} 3  7 42 7 52 43 43 4 7
S_{7}  12 1 7  5 43 43 32 6 5 5 6
2. 3_{i} 9_{+} 11_{k} -1 0 -1 -2 1^{4} 9_{+} 2∗ −1 1^{6} 11_{k} 3  7 5 42 43 43 52 43 5
S_{7}  0 4 2  3 0_{+} 1  42 6 6 43 43 322 7 43
3. 3_{i} 13_{−} 9_{i} -2 6 -10 14 1^{4} 11_{−} 2  2 1^{6} 9_{i} 3  5 7 7 43 52 52 6 7
S_{7}  -12 4 4  1^{3} 2_{+} ∗  7 6 43 6 6 42 42 32
4. 3_{i} 13_{+} 11_{k} 2 0 0 27 1^{4} 10_{i} −1  2 1^{6} 11_{k} 3  7 322 7 6 6 43 6 322
S_{7}  54 42 12  1^{2} 3_{k} −2∗ −2∗  33 52 6 6 32 33 33 32

Discriminant class: −3

∞  2  3 a_{1} a_{2} a_{3} a_{4}  Over p = 2  Over p = 3  5 7 11 13 17 19 23 29
G  h  a_{5} a_{6} a_{7}  f^{e} c_{w} d  s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
5. 3_{i} 12_{+} 11_{k} -1 0 0 18 1^{4} 10_{k} −1 2∗ 1^{6} 11_{k} 3∗  6 7 43 42 6 7 52 43
S_{7} -18 -24 -48 1^{2} 2_{i} −∗ −∗ 33 7 6 33 43 6 6 7

Discriminant class: −2

∞  2 3 a_{1} a_{2} a_{3} a_{4}  Over p = 2  Over p = 3 5 7 11 13 17 19 23 29
G  h  a_{5} a_{6} a_{7}  f^{e} c_{w} d  s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
6. 3_{i} 11_{k} 8_{+} 0 -3 1 3 1^{4} 11_{k} −2 −2∗ 2^{3} 8_{+} 1  6 52 42 52 7 7 43 43
S_{7} -6 -1 9 3 0_{+} 1 43 6 7 42 6 6 5 6

Discriminant class: −1

∞  2  3 a_{1} a_{2} a_{3} a_{4}  Over p = 2  Over p = 3 5 7 11 13 17 19 23 29
G  h  a_{5} a_{6} a_{7}  f^{e} c_{w} d  s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
7. 3_{i} 10_{k} 10_{+} -2 3 1 -5 1^{4} 10_{k} −1 2∗ 2^{3} 10_{+} ∗  42 43 6 322 33 32 6 42
S_{7}  12 -7 5  3 0_{+} 1  43 33 42 52 43 7 6 7
8. 3_{i} 10_{k} 10_{+} 0 -3 2 3 1^{6} 10_{k} −1  2^{3} 10_{+} ∗  7 43 6 5 7 43 43 5
S_{7}  -12 19 -6  52 42 7 4 6 33 6 322
9. 3_{i} 12_{k} 10_{+} -2 3 -4 5 1^{4} 9_{+} 2∗ −1 1^{6} 10_{+} ∗  7 43 6 7 42 52 6 7
S_{7} -6 7 -8 1^{2} 3_{k} −2∗ −2∗ 43 42 7 43 6 7 52 5
10. 3_{i} 12_{k} 10_{+} -3 3 1 -3 1^{4} 10_{k} −∗ −2∗ 1^{6} 10_{+} ∗  7 6 6 7 42 6 32 322
S_{7}  -3 -5 -3  1^{3} 2_{+} ∗  6 42 33 52 6 7 6 7