Low Degree Fields Unramified Outside of {∞, 2, 5, 7}

Discriminant class: −70

∞ 2 5 7 a_{1} a_{2} a_{3} Over p = 2 Over p = 5 Over p = 7 3 11 13 17 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_{+} 1_{k} 0 70 1^{2} 3_{−} 2∗ 2∗ 1^{2} 1_{+} 5 5 1^{2} 1_{k} 7 7 2 2 2 2 2 2
C_{2} 4  2

Discriminant class: −35

∞ 2 5 7 a_{1} a_{2} a_{3} Over p = 2 Over p = 5 Over p = 7 3 11 13 17 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_{−} 1_{k} 0 35  2 0_{+} ∗ ∗ 1^{2} 1_{−} 5∗ 5∗ 1^{2} 1_{k} 7 7  2 2  2
C_{2} 2  2 2 2  2 2 2 2
3. 1_{k} 2_{+} 1_{−} 1_{k} 0 2 -2 1^{3} 2_{+} ∗  1^{2} 1_{−} 5∗ 5∗ 1^{2} 1_{k} 7 7 3 3 3 3 2 2 3 2
S_{3}  2 2 2 3 2 2 2 2

Discriminant class: −14

∞ 2 5 7 a_{1} a_{2} a_{3} Over p = 2 Over p = 5 Over p = 7 3 11 13 17 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_{+} 0_{+} 1_{i} 0 14  1^{2} 3_{+} 2 2  1^{2} 1_{i} 7∗ 7∗  2  2  2 2
C_{2} 4  2 2 2 2 2  2

Discriminant class: −10

∞ 2 5 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  Over p = 7 3 11 13 17 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} 3_{k} 1_{−} 0_{+} 0 10  1^{2} 3_{k} −2∗ −2∗ 1^{2} 1_{−} 5∗ 5∗  2  2  2 2
C_{2} 2  2  2 2
6. 1_{k} 3_{k} 1_{−} 2_{+} -1 5 15 1^{2} 3_{k} −2∗ −2∗ 1^{2} 1_{−} 5∗ 5∗ 1^{3} 2_{+} 1  2 3 3 2 3 3 2 2
S_{3} 3 3 3 2 3 3 2 2

Discriminant class: −7

∞ 2 5 7 a_{1} a_{2} a_{3} Over p = 2 Over p = 5 Over p = 7 3 11 13 17 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} 0_{+} 0_{+} 1_{i} 1 2  2 0_{+} ∗ ∗ 1^{2} 1_{i} 7∗ 7∗ 2  2 2 2  2
C_{2}  2  2  2 2
8. 1_{k} 0_{+} 2_{+} 1_{i} -1 2 -3 3 0_{+} 1  1^{3} 2_{+} ∗  1^{2} 1_{i} 7∗ 7∗ 2 3 2 2 2 3 3 2
S_{3}  3 2 3 2  2 2 3

Discriminant class: −5

∞ 2 5 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  Over p = 7 3 11 13 17 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 2 2 2  2
C_{2} 2  2  2 2
10. 1_{k} 2_{i} 1_{+} 2_{+} -1 5 -13 1^{2} 2_{i} −∗ −∗ 1^{2} 1_{+} 5 5 1^{3} 2_{+} 1  3 2 2 2 2 3 3 2
S_{3} 3  2 3 3  2 2 3 3

Discriminant class: −2

∞ 2 5 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  Over p = 7 3 11 13 17 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 0_{+} ∗ ∗  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_{+} ∗  2 0_{+} ∗ ∗ 3 3 2 3 3 2 2 2
S_{3}  2 3  2 2  2 3

Discriminant class: −1

∞ 2 5 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  Over p = 7 3 11 13 17 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 0_{+} ∗ ∗ 2 2  2 2  2
C_{2}  2 2  2  2

Discriminant class: 1

∞ 2 5 7 a_{1} a_{2} a_{3} Over p = 2  Over p = 5  Over p = 7 3 11 13 17 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_{+} 0_{+} 0_{+} 2_{+} -1 -2 1 3 0_{+} 1  3 0_{+} 1  1^{3} 2_{+} 1  3 3  3 3 3  3
C_{3}  3  3 3 3 3 3

Discriminant class: 2

∞ 2 5 7 a_{1} a_{2} a_{3} Over p = 2  Over p = 5  Over p = 7 3 11 13 17 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_{+} 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
16. 0_{+} 3_{+} 2_{+} 2_{+} -1 -23 -13 1^{2} 3_{+} 2 2 1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2 2 2 3 2 3 2 3
S_{3} 3  2 3 2 3 2 2 2 2

Discriminant class: 5

∞ 2 5 7 a_{1} a_{2} a_{3} Over p = 2  Over p = 5  Over p = 7 3 11 13 17 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_{+} 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

Discriminant class: 7

∞ 2 5 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  Over p = 7 3 11 13 17 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_{+} 2_{i} 0_{+} 1_{k} 0 -7  1^{2} 2_{i} −1 −1 2 0_{+} ∗ ∗ 1^{2} 1_{k} 7 7  2 2 2  2
C_{2}  2 2  2 2

Discriminant class: 10

∞ 2 5 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  Over p = 7 3 11 13 17 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_{−} 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: 14

∞ 2 5 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  Over p = 7 3 11 13 17 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_{+} 3_{i} 0_{+} 1_{k} 0 -14  1^{2} 3_{i} −2 −2  1^{2} 1_{k} 7 7 2  2 2 2 2
C_{2}  2 2  2 2

Discriminant class: 35

∞ 2 5 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  Over p = 7  3 11 13 17 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_{+} 2_{i} 1_{−} 1_{i} 0 -35  1^{2} 2_{i} −∗ −∗ 1^{2} 1_{−} 5∗ 5∗ 1^{2} 1_{i} 7∗ 7∗ 2 2
C_{2} 2  2 2  2 2  2

Discriminant class: 70

∞ 2 5 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  Over p = 7  3 11 13 17 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
22. 0_{+} 3_{k} 1_{+} 1_{i} 0 -70  1^{2} 3_{k} −2∗ −2∗ 1^{2} 1_{+} 5 5 1^{2} 1_{i} 7∗ 7∗  2  2  2
C_{2} 2  2 2 2  2  2