Low Degree Fields Unramified Outside of {∞, 2, 17, 19}

Discriminant class: −646

∞ 2 17 19 a_{1} a_{2} a_{3} Over p = 2 Over p = 17 Over p = 19 3 5 7 11 13 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} 0 646  1^{2} 3_{−} 2∗ 2∗ 1^{2} 1_{+} 17 17 1^{2} 1_{k} 19 19 2  2  2 2
C_{2} 16  2  2

Discriminant class: −323

∞ 2 17 19 a_{1} a_{2} a_{3} Over p = 2  Over p = 17  Over p = 19  3 5 7 11 13 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_{i} 0 323  2 0_{+} ∗ ∗ 1^{2} 1_{+} 17 17 1^{2} 1_{i} 19∗ 19∗  2 2 2 2 2
C_{2} 4  2 2 2 2
3. 1_{k} 2_{+} 1_{+} 1_{i} 0 -10 -14 1^{3} 2_{+} ∗  1^{2} 1_{+} 17 17 1^{2} 1_{i} 19∗ 19∗ 3 2 2 2 2 2 3 3
S_{3}  3 3 2 2 2 2

Discriminant class: −38

∞ 2 17 19 a_{1} a_{2} a_{3} Over p = 2 Over p = 17 Over p = 19 3 5 7 11 13 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_{k} 0 38  1^{2} 3_{−} 2∗ 2∗  1^{2} 1_{k} 19 19  2  2  2
C_{2} 6  2 2  2
5. 1_{k} 3_{−} 0_{+} 1_{k} -1 -2 -2 1^{2} 3_{−} 2∗ 2∗ 3 0_{+} 1  1^{2} 1_{k} 19 19 3 2 3 2 3 3 3 2
S_{3} 2 2 3 3 2 3

Discriminant class: −34

∞ 2 17 19 a_{1} a_{2} a_{3}  Over p = 2  Over p = 17  Over p = 19 3 5 7 11 13 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} 3_{i} 1_{+} 0_{+} 0 34  1^{2} 3_{i} −2 −2 1^{2} 1_{+} 17 17  2  2 2
C_{2} 4  2  2 2
7. 1_{k} 3_{i} 1_{+} 2_{+} -1 -6 -126 1^{2} 3_{i} −2 −2 1^{2} 1_{+} 17 17 1^{3} 2_{+} 1  2 3  2 2 3 3 3
S_{3} 9  3 2 3 2 2 3

Discriminant class: −19

∞ 2 17 19 a_{1} a_{2} a_{3} Over p = 2  Over p = 17  Over p = 19  3 5 7 11 13 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} 0_{+} 0_{+} 1_{i} -1 5  2 0_{+} ∗ ∗  1^{2} 1_{i} 19∗ 19∗ 2  2  2 2
C_{2}  2 2  2 2  2
9. 1_{k} 2_{+} 0_{+} 1_{i} 0 -2 -2 1^{3} 2_{+} ∗  3 0_{+} 1  1^{2} 1_{i} 19∗ 19∗ 2 3 3 3 2  2 2
S_{3}  2 2 3 3 2 2 3 2

Discriminant class: −17

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

Discriminant class: −2

∞ 2 17 19 a_{1} a_{2} a_{3}  Over p = 2  Over p = 17 Over p = 19 3 5 7 11 13 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 2  2 2 2 2
C_{2}  2  2 2  2
12. 1_{k} 3_{i} 0_{+} 2_{+} -1 13 7 1^{2} 3_{i} −2 −2  1^{3} 2_{+} 1  3 2 2 3 2 2 2 2
S_{3} 3  2 3  2 2 3 2 3

Discriminant class: −1

∞ 2 17 19 a_{1} a_{2} a_{3}  Over p = 2  Over p = 17 Over p = 19 3 5 7 11 13 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 17 19 a_{1} a_{2} a_{3} Over p = 2  Over p = 17 Over p = 19 3 5 7 11 13 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 -6 7 3 0_{+} 1 3 0_{+} 1 1^{3} 2_{+} 1 3 3 3 3 3
C_{3}  3 3 3 3 3 3 3

Discriminant class: 2

∞ 2 17 19 a_{1} a_{2} a_{3} Over p = 2  Over p = 17 Over p = 19 3 5 7 11 13 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

Discriminant class: 17

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

Discriminant class: 19

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

Discriminant class: 34

∞ 2 17 19 a_{1} a_{2} a_{3} Over p = 2  Over p = 17  Over p = 19 3 5 7 11 13 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_{+} 3_{+} 1_{+} 0_{+} 0 -34  1^{2} 3_{+} 2 2 1^{2} 1_{+} 17 17 2 0_{+} ∗ ∗  2  2 2  2
C_{2} 2  2 2  2 2  2

Discriminant class: 38

∞ 2 17 19 a_{1} a_{2} a_{3}  Over p = 2  Over p = 17  Over p = 19  3 5 7 11 13 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_{k} 0_{+} 1_{i} 0 -38  1^{2} 3_{k} −2∗ −2∗  1^{2} 1_{i} 19∗ 19∗ 2 2 2  2
C_{2}  2  2  2 2 2

Discriminant class: 323

∞ 2 17 19 a_{1} a_{2} a_{3}  Over p = 2  Over p = 17  Over p = 19 3 5 7 11 13 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_{+} 1_{k} 0 -323  1^{2} 2_{i} −∗ −∗ 1^{2} 1_{+} 17 17 1^{2} 1_{k} 19 19 2 2  2  2
C_{2} 4  2 2 2  2

Discriminant class: 646

∞ 2 17 19 a_{1} a_{2} a_{3}  Over p = 2  Over p = 17  Over p = 19  3 5 7 11 13 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_{k} 1_{+} 1_{i} 0 -646  1^{2} 3_{k} −2∗ −2∗ 1^{2} 1_{+} 17 17 1^{2} 1_{i} 19∗ 19∗  2  2 2
C_{2} 8  2  2  2  2