Low Degree Fields Unramified Outside of {∞, 2, 13, 17}

Discriminant class: −442

∞ 2 13 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 13  Over p = 17 3 5 7 11 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_{+} 0 442  1^{2} 3_{k} −2∗ −2∗ 1^{2} 1_{−} 13∗ 13∗ 1^{2} 1_{+} 17 17 2 2 2  2  2
C_{2} 8  2  2 2  2

Discriminant class: −221

∞ 2 13 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 13  Over p = 17 3 5 7 11 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} 2_{i} 1_{+} 1_{+} 0 221  1^{2} 2_{i} −∗ −∗ 1^{2} 1_{+} 13 13 1^{2} 1_{+} 17 17  2 2  2 2
C_{2} 16  2  2

Discriminant class: −34

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

Discriminant class: −26

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

Discriminant class: −17

∞ 2 13 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 13  Over p = 17 3 5 7 11 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
6. 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
7. 1_{k} 2_{i} 2_{+} 1_{+} -1 -4 -40 1^{2} 2_{i} −1 −1 1^{3} 2_{+} 1  1^{2} 1_{+} 17 17 3 2 3  2 3 2 3
S_{3} 3  2 2 2 2 3 2 2 2

Discriminant class: −13

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

Discriminant class: −2

∞ 2 13 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 13 Over p = 17 3 5 7 11 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} 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 13 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 13 Over p = 17 3 5 7 11 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
10. 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
11. 1_{k} 2_{i} 2_{+} 0_{+} -1 -4 12 1^{2} 2_{i} −1 −1 1^{3} 2_{+} 1  3 0_{+} 1  2 3 2 2 2 2 3 2
S_{3} 3  3 3 2 2 3 2 3 2

Discriminant class: 1

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

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

Discriminant class: 26

∞ 2 13 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 13  Over p = 17 3 5 7 11 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
16. 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: 34

∞ 2 13 17 a_{1} a_{2} a_{3} Over p = 2  Over p = 13  Over p = 17 3 5 7 11 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_{+} 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: 221

∞ 2 13 17 a_{1} a_{2} a_{3} Over p = 2  Over p = 13  Over p = 17 3 5 7 11 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_{+} 1_{+} 1_{+} 0 -221  2 0_{+} ∗ ∗ 1^{2} 1_{+} 13 13 1^{2} 1_{+} 17 17 2  2 2 2
C_{2} 2  2  2 2 2

Discriminant class: 442

∞ 2 13 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 13  Over p = 17 3 5 7 11 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_{−} 1_{+} 0 -442  1^{2} 3_{−} 2∗ 2∗ 1^{2} 1_{−} 13∗ 13∗ 1^{2} 1_{+} 17 17  2  2  2
C_{2} 8  2  2 2 2