Low Degree Fields Unramified Outside of {∞, 2, 5, 17}

Discriminant class: −170

∞ 2 5 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  Over p = 17  3 7 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_{k} 1_{+} 1_{−} 0 170  1^{2} 3_{k} −2∗ −2∗ 1^{2} 1_{+} 5 5 1^{2} 1_{−} 17∗ 17∗  2 2  2
C_{2} 12  2 2 2  2
2. 1_{k} 3_{k} 1_{+} 1_{−} -1 -6 10 1^{2} 3_{k} −2∗ −2∗ 1^{2} 1_{+} 5 5 1^{2} 1_{−} 17∗ 17∗ 3 2 2 3 3 2 3 3
S_{3}  2 2 2 3 3 3 3 2

Discriminant class: −85

∞ 2 5 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  Over p = 17  3 7 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} 2_{i} 1_{−} 1_{−} 0 85  1^{2} 2_{i} −∗ −∗ 1^{2} 1_{−} 5∗ 5∗ 1^{2} 1_{−} 17∗ 17∗ 2 2  2 2 2 2
C_{2} 4  2  2 2 2

Discriminant class: −34

∞ 2 5 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  Over p = 17 3 7 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
4. 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

Discriminant class: −17

∞ 2 5 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  Over p = 17 3 7 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 2 0_{+} ∗ ∗ 1^{2} 1_{+} 17 17  2  2
C_{2} 4  2 2 2 2  2 2 2
6. 1_{k} 2_{i} 2_{+} 1_{+} -1 12 -8 1^{2} 2_{i} −1 −1 1^{3} 2_{+} ∗  1^{2} 1_{+} 17 17 3 3  3 2  2 3
S_{3}  2 2 2 2 3 2 2 2

Discriminant class: −10

∞ 2 5 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  Over p = 17 3 7 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_{k} 1_{−} 0_{+} 0 10  1^{2} 3_{k} −2∗ −2∗ 1^{2} 1_{−} 5∗ 5∗ 2 0_{+} ∗ ∗ 2  2 2
C_{2} 2  2  2 2
8. 1_{k} 3_{k} 1_{−} 2_{+} 0 -17 -34 1^{2} 3_{k} −2∗ −2∗ 1^{2} 1_{−} 5∗ 5∗ 1^{3} 2_{+} ∗  2 3 3 3  3 2 2
S_{3}  2 3 3 3 2 2

Discriminant class: −5

∞ 2 5 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  Over p = 17 3 7 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} 2_{i} 1_{+} 0_{+} 0 5  1^{2} 2_{i} −∗ −∗ 1^{2} 1_{+} 5 5 2 0_{+} ∗ ∗  2 2 2  2
C_{2} 2  2  2 2
10. 1_{k} 2_{i} 1_{+} 2_{+} -1 -11 -29 1^{2} 2_{i} −∗ −∗ 1^{2} 1_{+} 5 5 1^{3} 2_{+} ∗  3 3 2 2 2 3  2
S_{3}  2 3  2 2  3

Discriminant class: −2

∞ 2 5 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  Over p = 17 3 7 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
12. 1_{k} 3_{i} 2_{+} 0_{+} -1 2 2 1^{2} 3_{i} −2 −2 1^{3} 2_{+} ∗  3 0_{+} 1  3 2 3 2 3 2 2 2
S_{3}  2 3  2 2  2 3

Discriminant class: −1

∞ 2 5 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  Over p = 17 3 7 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. 1_{k} 2_{i} 0_{+} 0_{+} 0 1  1^{2} 2_{i} −1 −1  2 2 2  2 2  2
C_{2}  2 2  2  2

Discriminant class: 2

∞ 2 5 17 a_{1} a_{2} a_{3} Over p = 2  Over p = 5  Over p = 17 3 7 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 0_{+} ∗ ∗  2  2 2 2  2
C_{2}  2  2  2 2 2 2

Discriminant class: 5

∞ 2 5 17 a_{1} a_{2} a_{3} Over p = 2  Over p = 5  Over p = 17 3 7 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_{+} 0_{+} 1_{+} 0_{+} 1 -1  2 0_{+} ∗ ∗ 1^{2} 1_{+} 5 5 2 0_{+} ∗ ∗ 2 2  2  2
C_{2}  2  2 2 2  2
16. 0_{+} 2_{+} 1_{+} 2_{+} -1 -11 5 1^{3} 2_{+} ∗  1^{2} 1_{+} 5 5 1^{3} 2_{+} ∗  2 2 3 2 3 2 3
S_{3}  2 3 2 2 2 3 3 2

Discriminant class: 10

∞ 2 5 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  Over p = 17 3 7 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_{−} 1_{−} 0_{+} 0 -10  1^{2} 3_{−} 2∗ 2∗ 1^{2} 1_{−} 5∗ 5∗ 2 0_{+} ∗ ∗  2 2  2 2 2
C_{2} 2  2  2 2

Discriminant class: 17

∞ 2 5 17 a_{1} a_{2} a_{3} Over p = 2  Over p = 5  Over p = 17 3 7 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 5 17 a_{1} a_{2} a_{3} Over p = 2  Over p = 5  Over p = 17 3 7 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  1^{2} 1_{+} 17 17  2  2 2 2  2
C_{2} 2  2 2  2 2  2

Discriminant class: 85

∞ 2 5 17 a_{1} a_{2} a_{3} Over p = 2  Over p = 5  Over p = 17  3 7 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_{+} 0_{+} 1_{−} 1_{−} 0 -85  2 0_{+} ∗ ∗ 1^{2} 1_{−} 5∗ 5∗ 1^{2} 1_{−} 17∗ 17∗  2 2  2 2
C_{2} 2  2 2 2 2  2 2

Discriminant class: 170

∞ 2 5 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  Over p = 17  3 7 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_{−} 1_{+} 1_{−} 0 -170  1^{2} 3_{−} 2∗ 2∗ 1^{2} 1_{+} 5 5 1^{2} 1_{−} 17∗ 17∗ 2  2  2
C_{2} 4  2 2  2  2