Low Degree Fields Unramified Outside of {∞, 2, 7, 17}

Discriminant class: −238

∞ 2 7 17 a_{1} a_{2} a_{3} Over p = 2  Over p = 7  Over p = 17  3 5 11 13 19 23 29 31
G h  f^{e} c_{w} d s f^{e} c_{w} d s f^{e} c_{w} d  s 37 41 43 47 53 59 61 67
1. 1_{k} 3_{+} 1_{k} 1_{−} 0 238  1^{2} 3_{+} 2 2 1^{2} 1_{k} 7 7 1^{2} 1_{−} 17∗ 17∗ 2 2  2
C_{2} 8  2 2 2  2 2

Discriminant class: −119

∞ 2 7 17 a_{1} a_{2} a_{3} Over p = 2  Over p = 7  Over p = 17  3 5 11 13 19 23 29 31
G h  f^{e} c_{w} d s f^{e} c_{w} d s f^{e} c_{w} d  s 37 41 43 47 53 59 61 67
2. 1_{k} 0_{+} 1_{k} 1_{−} 0 119  1^{2} 1_{k} 7 7 1^{2} 1_{−} 17∗ 17∗  2 2 2 2 2
C_{2} 10  2  2  2

Discriminant class: −34

∞ 2 7 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 7  Over p = 17 3 5 11 13 19 23 29 31
G h  f^{e} c_{w} d  s f^{e} c_{w} d s f^{e} c_{w} d s 37 41 43 47 53 59 61 67
3. 1_{k} 3_{i} 0_{+} 1_{+} 0 34  1^{2} 3_{i} −2 −2  1^{2} 1_{+} 17 17 2  2 2
C_{2} 4  2  2 2
4. 1_{k} 3_{i} 2_{+} 1_{+} 0 7 -14 1^{2} 3_{i} −2 −2 1^{3} 2_{+} 1  1^{2} 1_{+} 17 17 2 3 2 2 3 3
S_{3} 6 3 2 3 2 2 3 3

Discriminant class: −17

∞ 2 7 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 7  Over p = 17 3 5 11 13 19 23 29 31
G h  f^{e} c_{w} d  s f^{e} c_{w} d s f^{e} c_{w} d s 37 41 43 47 53 59 61 67
5. 1_{k} 2_{i} 0_{+} 1_{+} 0 17  1^{2} 2_{i} −1 −1  1^{2} 1_{+} 17 17  2  2  2
C_{2} 4  2 2 2 2  2 2 2
6. 1_{k} 2_{i} 2_{+} 1_{+} -1 12 -20 1^{2} 2_{i} −1 −1 1^{3} 2_{+} 1  1^{2} 1_{+} 17 17 3 2 3 3 2  2
S_{3} 3  2 2 2 2 3 2 2 2

Discriminant class: −14

∞ 2 7 17 a_{1} a_{2} a_{3} Over p = 2 Over p = 7 Over p = 17 3 5 11 13 19 23 29 31
G h  f^{e} c_{w} d s f^{e} c_{w} d s f^{e} c_{w} d s 37 41 43 47 53 59 61 67
7. 1_{k} 3_{+} 1_{i} 0_{+} 0 14  1^{2} 3_{+} 2 2 1^{2} 1_{i} 7∗ 7∗ 2 0_{+} ∗ ∗  2  2 2
C_{2} 4  2 2 2 2 2  2
8. 1_{k} 3_{+} 1_{i} 2_{+} -1 6 22 1^{2} 3_{+} 2 2 1^{2} 1_{i} 7∗ 7∗ 1^{3} 2_{+} ∗  3 3 2 3  3 2 2
S_{3}  2 2 2 2 2  3 2

Discriminant class: −7

∞ 2 7 17 a_{1} a_{2} a_{3} Over p = 2  Over p = 7  Over p = 17 3 5 11 13 19 23 29 31
G h  f^{e} c_{w} d s f^{e} c_{w} d s f^{e} c_{w} d s 37 41 43 47 53 59 61 67
9. 1_{k} 0_{+} 1_{i} 0_{+} 1 2  1^{2} 1_{i} 7∗ 7∗ 2 0_{+} ∗ ∗ 2 2  2 2  2
C_{2}  2  2  2 2
10. 1_{k} 0_{+} 1_{i} 2_{+} -1 6 5 3 0_{+} 1  1^{2} 1_{i} 7∗ 7∗ 1^{3} 2_{+} ∗  2 2  2 2 3 3 2
S_{3}  3 2 3 2 3 2 2

Discriminant class: −2

∞ 2 7 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 7  Over p = 17 3 5 11 13 19 23 29 31
G h  f^{e} c_{w} d  s f^{e} c_{w} d s f^{e} c_{w} d s 37 41 43 47 53 59 61 67
11. 1_{k} 3_{i} 0_{+} 0_{+} 0 2  1^{2} 3_{i} −2 −2 2 0_{+} ∗ ∗  2  2  2 2 2
C_{2}  2  2 2  2

Discriminant class: −1

∞ 2 7 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 7  Over p = 17 3 5 11 13 19 23 29 31
G h  f^{e} c_{w} d  s f^{e} c_{w} d s f^{e} c_{w} d s 37 41 43 47 53 59 61 67
12. 1_{k} 2_{i} 0_{+} 0_{+} 0 1  1^{2} 2_{i} −1 −1 2 0_{+} ∗ ∗  2  2  2 2  2
C_{2}  2 2  2  2

Discriminant class: 1

∞ 2 7 17 a_{1} a_{2} a_{3} Over p = 2  Over p = 7  Over p = 17 3 5 11 13 19 23 29 31
G h  f^{e} c_{w} d s f^{e} c_{w} d s f^{e} c_{w} d s 37 41 43 47 53 59 61 67
13. 0_{+} 0_{+} 2_{+} 0_{+} -1 -2 1 3 0_{+} 1  1^{3} 2_{+} 1  3 0_{+} 1  3 3 3  3 3  3
C_{3}  3  3 3 3 3 3

Discriminant class: 2

∞ 2 7 17 a_{1} a_{2} a_{3} Over p = 2  Over p = 7  Over p = 17 3 5 11 13 19 23 29 31
G h  f^{e} c_{w} d s f^{e} c_{w} d s f^{e} c_{w} d s 37 41 43 47 53 59 61 67
14. 0_{+} 3_{+} 0_{+} 0_{+} 0 -2  1^{2} 3_{+} 2 2  2 2 2 2 2  2
C_{2}  2  2  2 2 2 2

Discriminant class: 7

∞ 2 7 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 7  Over p = 17 3 5 11 13 19 23 29 31
G h  f^{e} c_{w} d  s f^{e} c_{w} d s f^{e} c_{w} d s 37 41 43 47 53 59 61 67
15. 0_{+} 2_{i} 1_{k} 0_{+} 0 -7  1^{2} 2_{i} −1 −1 1^{2} 1_{k} 7 7 2 0_{+} ∗ ∗  2 2 2  2
C_{2}  2 2  2 2
16. 0_{+} 2_{i} 1_{k} 2_{+} -1 -28 -46 1^{2} 2_{i} −1 −1 1^{2} 1_{k} 7 7 1^{3} 2_{+} ∗  3 2 2 2 3 2 3 3
S_{3}  2 2 3 3 3 2 2

Discriminant class: 14

∞ 2 7 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 7  Over p = 17 3 5 11 13 19 23 29 31
G h  f^{e} c_{w} d  s f^{e} c_{w} d s f^{e} c_{w} d s 37 41 43 47 53 59 61 67
17. 0_{+} 3_{i} 1_{k} 0_{+} 0 -14  1^{2} 3_{i} −2 −2 1^{2} 1_{k} 7 7 2 0_{+} ∗ ∗ 2  2 2 2
C_{2}  2 2  2 2

Discriminant class: 17

∞ 2 7 17 a_{1} a_{2} a_{3} Over p = 2  Over p = 7  Over p = 17 3 5 11 13 19 23 29 31
G h  f^{e} c_{w} d s f^{e} c_{w} d s f^{e} c_{w} d s 37 41 43 47 53 59 61 67
18. 0_{+} 0_{+} 0_{+} 1_{+} 1 -4  2 0_{+} ∗ ∗ 1^{2} 1_{+} 17 17 2 2 2  2 2 2
C_{2}  2 2  2

Discriminant class: 34

∞ 2 7 17 a_{1} a_{2} a_{3} Over p = 2  Over p = 7  Over p = 17 3 5 11 13 19 23 29 31
G h  f^{e} c_{w} d s f^{e} c_{w} d s f^{e} c_{w} d s 37 41 43 47 53 59 61 67
19. 0_{+} 3_{+} 0_{+} 1_{+} 0 -34  1^{2} 3_{+} 2 2 2 0_{+} ∗ ∗ 1^{2} 1_{+} 17 17  2 2 2  2
C_{2} 2  2 2  2 2  2

Discriminant class: 119

∞ 2 7 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 7  Over p = 17  3 5 11 13 19 23 29 31
G h  f^{e} c_{w} d  s f^{e} c_{w} d s f^{e} c_{w} d  s 37 41 43 47 53 59 61 67
20. 0_{+} 2_{i} 1_{i} 1_{−} 0 -119  1^{2} 2_{i} −1 −1 1^{2} 1_{i} 7∗ 7∗ 1^{2} 1_{−} 17∗ 17∗ 2  2  2 2
C_{2} 2  2  2  2

Discriminant class: 238

∞ 2 7 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 7  Over p = 17  3 5 11 13 19 23 29 31
G h  f^{e} c_{w} d  s f^{e} c_{w} d s f^{e} c_{w} d  s 37 41 43 47 53 59 61 67
21. 0_{+} 3_{i} 1_{i} 1_{−} 0 -238  1^{2} 3_{i} −2 −2 1^{2} 1_{i} 7∗ 7∗ 1^{2} 1_{−} 17∗ 17∗  2 2  2  2
C_{2} 2  2 2 2