Low Degree Fields Unramified Outside of {∞, 2, 13, 23}

Discriminant class: −598

∞ 2 13 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 13  Over p = 23  3 5 7 11 17 19 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_{i} 0 598  1^{2} 3_{−} 2∗ 2∗ 1^{2} 1_{−} 13∗ 13∗ 1^{2} 1_{i} 23∗ 23∗ 2 2  2 2 2 2 2
C_{2} 8  2 2  2  2

Discriminant class: −299

∞ 2 13 23 a_{1} a_{2} a_{3} Over p = 2  Over p = 13  Over p = 23  3 5 7 11 17 19 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_{i} 0 299  2 0_{+} ∗ ∗ 1^{2} 1_{+} 13 13 1^{2} 1_{i} 23∗ 23∗  2  2
C_{2} 8  2 2 2 2 2 2
3. 1_{k} 2_{+} 1_{+} 1_{i} -1 5 -7 1^{3} 2_{+} ∗ 1^{2} 1_{+} 13 13 1^{2} 1_{i} 23∗ 23∗ 3 3 3 2 3 2
S_{3}  2 2 2 2 2 2 3

Discriminant class: −46

∞ 2 13 23 a_{1} a_{2} a_{3} Over p = 2  Over p = 13  Over p = 23  3 5 7 11 17 19 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 46  1^{2} 3_{+} 2 2 2 0_{+} ∗ ∗ 1^{2} 1_{i} 23∗ 23∗ 2  2  2  2
C_{2} 4  2

Discriminant class: −26

∞ 2 13 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 13  Over p = 23 3 5 7 11 17 19 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 26  1^{2} 3_{k} −2∗ −2∗ 1^{2} 1_{−} 13∗ 13∗ 2 0_{+} ∗ ∗  2  2 2
C_{2} 6  2  2 2 2 2
6. 1_{k} 3_{k} 1_{−} 0_{+} 0 -1 -2 1^{2} 3_{k} −2∗ −2∗ 1^{2} 1_{−} 13∗ 13∗ 2 0_{+} ∗ ∗ 3 3 3 2 3 2 2
S_{3}  3 2 3 3 2 2 2 2

Discriminant class: −23

∞ 2 13 23 a_{1} a_{2} a_{3} Over p = 2  Over p = 13  Over p = 23  3 5 7 11 17 19 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} 0 23  1^{2} 1_{i} 23∗ 23∗  2 2 2 2 2
C_{2} 3  2  2  2  2 2
8. 1_{k} 0_{+} 0_{+} 1_{i} -1 0 1 3 0_{+} 1  3 0_{+} 1  1^{2} 1_{i} 23∗ 23∗ 3 2 2 2 2 2 3 3
S_{3}  2 3 2 3 2  2 2

Discriminant class: −13

∞ 2 13 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 13  Over p = 23 3 5 7 11 17 19 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 13  1^{2} 2_{i} −∗ −∗ 1^{2} 1_{+} 13 13 2 0_{+} ∗ ∗ 2 2
C_{2} 2  2 2 2
10. 1_{k} 2_{i} 1_{+} 2_{+} -1 8 -98 1^{2} 2_{i} −∗ −∗ 1^{2} 1_{+} 13 13 1^{3} 2_{+} ∗  2 2  3 3 3 3 3
S_{3}  2 2 2 3 3 3  3

Discriminant class: −2

∞ 2 13 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 13 Over p = 23 3 5 7 11 17 19 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} 0_{+} 2_{+} -1 -15 -29 1^{2} 3_{i} −2 −2 2 0_{+} ∗ ∗ 1^{3} 2_{+} ∗  3 2 2  3 3 2 2
S_{3}  2  3 2 2 3 2 3

Discriminant class: −1

∞ 2 13 23 a_{1} a_{2}  a_{3}  Over p = 2  Over p = 13 Over p = 23 3 5 7 11 17 19 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} 2_{+} 0_{+} -1 -4  12 1^{2} 2_{i} −1 −1 1^{3} 2_{+} 1  2 0_{+} ∗ ∗ 2 3 2 2 3 2 3 2
S_{3} 3  3 3 2 2 3 2 3 2
15. 1_{k} 2_{i} 0_{+} 2_{+} -1 8 -6 1^{2} 2_{i} −1 −1 3 0_{+} 1 1^{3} 2_{+} ∗ 2 3 2 2 2 3 2
S_{3} 2  3 2 2 3 2 3 2
16. 1_{k} 2_{i} 2_{+} 2_{+} -1 100 -742 1^{2} 2_{i} −1 −1 1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 3 2 2 3 2  2
S_{3} 3  3 3 2 2  2  2
17. 1_{k} 2_{i} 2_{+} 2_{+} 1 100 -1052 1^{2} 2_{i} −1 −1 1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2  2 2 3 2 3 2
S_{3} 6  3  2 2 3 2 3 2

Discriminant class: 1

∞ 2 13 23 a_{1} a_{2} a_{3} Over p = 2  Over p = 13 Over p = 23 3 5 7 11 17 19 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_{+} 2_{+} 0_{+} -1 -4 -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 13 23 a_{1} a_{2} a_{3} Over p = 2  Over p = 13 Over p = 23 3 5 7 11 17 19 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: 13

∞ 2 13 23 a_{1} a_{2} a_{3} Over p = 2  Over p = 13  Over p = 23 3 5 7 11 17 19 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_{+} 0_{+} 1 -3  2 0_{+} ∗ ∗ 1^{2} 1_{+} 13 13  2 2 2  2  2
C_{2}  2 2  2  2  2

Discriminant class: 23

∞ 2 13 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 13  Over p = 23 3 5 7 11 17 19 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} 0_{+} 1_{k} 0 -23  1^{2} 2_{i} −1 −1  1^{2} 1_{k} 23 23 2 2  2  2
C_{2}  2  2 2 2 2

Discriminant class: 26

∞ 2 13 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 13  Over p = 23 3 5 7 11 17 19 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_{−} 1_{−} 0_{+} 0 -26  1^{2} 3_{−} 2∗ 2∗ 1^{2} 1_{−} 13∗ 13∗  2  2  2 2
C_{2} 2  2 2 2 2  2

Discriminant class: 46

∞ 2 13 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 13  Over p = 23 3 5 7 11 17 19 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_{i} 0_{+} 1_{k} 0 -46  1^{2} 3_{i} −2 −2 2 0_{+} ∗ ∗ 1^{2} 1_{k} 23 23  2 2 2 2 2
C_{2}  2 2  2

Discriminant class: 299

∞ 2 13 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 13  Over p = 23 3 5 7 11 17 19 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_{+} 1_{k} 0 -299  1^{2} 2_{i} −∗ −∗ 1^{2} 1_{+} 13 13 1^{2} 1_{k} 23 23 2  2 2 2 2
C_{2} 2  2  2  2 2

Discriminant class: 598

∞ 2 13 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 13  Over p = 23 3 5 7 11 17 19 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_{−} 1_{k} 0 -598  1^{2} 3_{k} −2∗ −2∗ 1^{2} 1_{−} 13∗ 13∗ 1^{2} 1_{k} 23 23  2 2  2  2
C_{2} 2  2 2 2  2