Low Degree Fields Unramified Outside of {∞, 5, 13, 19}

Discriminant class: −1235

∞ 5 13 19 a_{1} a_{2} a_{3} Over p = 5 Over p = 13 Over p = 19 2 3 7 11 17 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} 1_{−} 1_{+} 1_{k} 0 1235  1^{2} 1_{−} 5∗ 5∗ 1^{2} 1_{+} 13 13 1^{2} 1_{k} 19 19 2  2 2 2 2
C_{2} 12  2  2  2
2. 1_{k} 1_{−} 1_{+} 1_{k} -1  -5 10 1^{2} 1_{−} 5∗ 5∗ 1^{2} 1_{+} 13 13 1^{2} 1_{k} 19 19 2 3 3 2 2 2 2 3
S_{3} 2 3 2 3 3 2

Discriminant class: −247

∞ 5 13 19 a_{1} a_{2} a_{3} Over p = 5  Over p = 13  Over p = 19 2 3 7 11 17 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_{−} 1_{k} 0 247  2 0_{+} ∗ ∗ 1^{2} 1_{−} 13∗ 13∗ 1^{2} 1_{k} 19 19  2 2 2  2
C_{2} 6  2 2
4. 1_{k} 0_{+} 1_{−} 1_{k} 0 1 -3 2 0_{+} ∗ ∗ 1^{2} 1_{−} 13∗ 13∗ 1^{2} 1_{k} 19 19 3 2 2 2 3 3 2 3
S_{3}  3 3 3 2 2 3 3 3

Discriminant class: −95

∞ 5 13 19 a_{1} a_{2} a_{3} Over p = 5  Over p = 13  Over p = 19  2 3 7 11 17 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} 1_{+} 0_{+} 1_{i} 0 95  1^{2} 1_{+} 5 5  1^{2} 1_{i} 19∗ 19∗  2  2 2 2 2
C_{2} 8  2 2 2  2
6. 1_{k} 1_{+} 2_{+} 1_{i} 0 -13 -52 1^{2} 1_{+} 5 5 1^{3} 2_{+} 1  1^{2} 1_{i} 19∗ 19∗  3 2 3 2 2 2 2
S_{3} 3  3 2 2 2 3 2 3 3

Discriminant class: −19

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

Discriminant class: 1

∞ 5 13 19 a_{1} a_{2} a_{3} Over p = 5  Over p = 13 Over p = 19 2 3 7 11 17 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. 0_{+} 0_{+} 2_{+} 0_{+} -1 -4 -1  1^{3} 2_{+} 1  3 0_{+} 1  3 3 3 3 3 3 3
C_{3}  3 3 3  3 3 3
9. 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
10. 0_{+} 0_{+} 2_{+} 2_{+} -1 -82 311 3 0_{+} 1  1^{3} 2_{+} 1  1^{3} 2_{+} 1  3 3 3 3
C_{3} 3  3 3  3 3  3

Discriminant class: 1 (continued)

∞ 5 13 19 a_{1} a_{2} a_{3} Over p = 5  Over p = 13 Over p = 19 2 3 7 11 17 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. 0_{+} 0_{+} 2_{+} 2_{+} -1 -82 64 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: 5

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

Discriminant class: 13

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

Discriminant class: 65

∞ 5 13 19 a_{1} a_{2} a_{3}  Over p = 5  Over p = 13  Over p = 19 2 3 7 11 17 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_{+} 1_{−} 1_{−} 0_{+} 0 -65  1^{2} 1_{−} 5∗ 5∗ 1^{2} 1_{−} 13∗ 13∗ 2 0_{+} ∗ ∗  2  2 2 2  2
C_{2} 2  2 2  2 2