Low Degree Fields Unramified Outside of {∞, 2, 7, 13}

Discriminant class: −182

∞ 2 7 13 a_{1} a_{2} a_{3} Over p = 2 Over p = 7 Over p = 13 3 5 11 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_{k} 1_{+} 0 182  1^{2} 3_{−} 2∗ 2∗ 1^{2} 1_{k} 7 7 1^{2} 1_{+} 13 13  2  2 2  2
C_{2} 12  2  2 2
2. 1_{k} 3_{−} 1_{k} 1_{+} -1 6 -2 1^{2} 3_{−} 2∗ 2∗ 1^{2} 1_{k} 7 7 1^{2} 1_{+} 13 13 3 2 3 2 2 3 2 3
S_{3}  3 3 2 3 2 2 3 3

Discriminant class: −91

∞ 2 7 13 a_{1} a_{2} a_{3} Over p = 2  Over p = 7  Over p = 13  3 5 11 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
3. 1_{k} 0_{+} 1_{k} 1_{−} 0 91  2 0_{+} ∗ ∗ 1^{2} 1_{k} 7 7 1^{2} 1_{−} 13∗ 13∗ 2  2 2
C_{2} 2  2  2 2
4. 1_{k} 2_{+} 1_{k} 1_{−} 0 4 -2 1^{3} 2_{+} ∗  1^{2} 1_{k} 7 7 1^{2} 1_{−} 13∗ 13∗ 2 3 2 2 3 3 3 3
S_{3} 2 3 3 3 2 2

Discriminant class: −26

∞ 2 7 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 7  Over p = 13  3 5 11 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} 0_{+} 1_{−} 0 26  1^{2} 3_{k} −2∗ −2∗  1^{2} 1_{−} 13∗ 13∗  2  2 2 2
C_{2} 6  2  2 2 2 2
6. 1_{k} 3_{k} 0_{+} 1_{−} 0 -1 -2 1^{2} 3_{k} −2∗ −2∗ 3 0_{+} 1  1^{2} 1_{−} 13∗ 13∗ 3 3 2 3 2 2 2
S_{3}  3 2 3 3 2 2 2 2

Discriminant class: −14

∞ 2 7 13 a_{1} a_{2} a_{3} Over p = 2 Over p = 7 Over p = 13 3 5 11 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} 3_{+} 1_{i} 0_{+} 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
8. 1_{k} 3_{+} 1_{i} 2_{+} 0 -13 -26 1^{2} 3_{+} 2 2 1^{2} 1_{i} 7∗ 7∗ 1^{3} 2_{+} 1  3 3 2 2  3 2 2
S_{3} 3  2 2 2 2 2 3 3 2

Discriminant class: −13

∞ 2 7 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 7  Over p = 13 3 5 11 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} 0_{+} 1_{+} 0 13  1^{2} 2_{i} −∗ −∗  1^{2} 1_{+} 13 13 2 2  2
C_{2} 2  2 2 2
10. 1_{k} 2_{i} 2_{+} 1_{+} -1 12 8 1^{2} 2_{i} −∗ −∗ 1^{3} 2_{+} 1  1^{2} 1_{+} 13 13 2 2 3 3 3 2 3
S_{3} 3  2 2 2 3 3 3 3 3

Discriminant class: −7

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

Discriminant class: −2

∞ 2 7 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 7  Over p = 13 3 5 11 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
12. 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

Discriminant class: −1

∞ 2 7 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 7  Over p = 13 3 5 11 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
14. 1_{k} 2_{i} 0_{+} 2_{+} -1 -4 12 1^{2} 2_{i} −1 −1 2 0_{+} ∗ ∗ 1^{3} 2_{+} 1  2 3 2 3 2 2 3 2
S_{3} 3  3 3 2 2 3 2 3 2

Discriminant class: 1

∞ 2 7 13 a_{1} a_{2} a_{3} Over p = 2  Over p = 7  Over p = 13 3 5 11 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_{+} 0_{+} 2_{+} 0_{+} -1 -2 1 3 0_{+} 1  1^{3} 2_{+} 1  3 3 3 3 3 3  3
C_{3}  3  3 3 3 3 3
16. 0_{+} 0_{+} 0_{+} 2_{+} -1 -4 -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
17. 0_{+} 0_{+} 2_{+} 2_{+} -1 -30 64  1^{3} 2_{+} 1  1^{3} 2_{+} 1  3 3 3  3  3 3
C_{3} 3  3 3 3 3  3 3
18. 0_{+} 0_{+} 2_{+} 2_{+} 1 -30 27 3 0_{+} 1  1^{3} 2_{+} 1  1^{3} 2_{+} 1  3  3  3 3 3
C_{3} 3  3 3 3 3 3 3

Discriminant class: 2

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

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

Discriminant class: 13

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

Discriminant class: 14

∞ 2 7 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 7  Over p = 13 3 5 11 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_{i} 1_{k} 0_{+} 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: 26

∞ 2 7 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 7  Over p = 13  3 5 11 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
23. 0_{+} 3_{−} 0_{+} 1_{−} 0 -26  1^{2} 3_{−} 2∗ 2∗ 2 0_{+} ∗ ∗ 1^{2} 1_{−} 13∗ 13∗ 2  2 2
C_{2} 2  2 2 2 2  2

Discriminant class: 91

∞ 2 7 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 7  Over p = 13  3 5 11 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
24. 0_{+} 2_{i} 1_{i} 1_{−} 0 -91  1^{2} 2_{i} −∗ −∗ 1^{2} 1_{i} 7∗ 7∗ 1^{2} 1_{−} 13∗ 13∗  2 2 2  2
C_{2} 2  2  2 2  2 2

Discriminant class: 182

∞ 2 7 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 7  Over p = 13 3 5 11 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
25. 0_{+} 3_{k} 1_{i} 1_{+} 0 -182  1^{2} 3_{k} −2∗ −2∗ 1^{2} 1_{i} 7∗ 7∗ 1^{2} 1_{+} 13 13 2 2 2 2  2 2 2
C_{2} 2  2 2  2