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

Discriminant class: −247

∞ 13 19 a_{1} a_{2} a_{3}  Over p = 13  Over p = 19 2 3 5 7 11 17 23 29
G h  f^{e} c_{w} d  s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
1. 1_{k} 1_{−} 1_{k} 0 247  1^{2} 1_{−} 13∗ 13∗ 1^{2} 1_{k} 19 19  2 2 2 2  2
C_{2} 6  2 2
2. 1_{k} 1_{−} 1_{k} 0 1 -3 1^{2} 1_{−} 13∗ 13∗ 1^{2} 1_{k} 19 19 3 2 2 2 2 3 3 2
S_{3}  3 3 3 3 2 2 3 3

Discriminant class: −19

∞ 13 19 a_{1} a_{2} a_{3} Over p = 13  Over p = 19  2 3 5 7 11 17 23 29
G h  f^{e} c_{w} d s f^{e} c_{w} d  s 31 37 41 43 47 53 59 61
3. 1_{k} 0_{+} 1_{i} -1 5  2 0_{+} ∗ ∗ 1^{2} 1_{i} 19∗ 19∗ 2 2  2
C_{2}  2 2 2  2 2

Discriminant class: 1

∞ 13 19 a_{1} a_{2} a_{3} Over p = 13 Over p = 19 2 3 5 7 11 17 23 29
G h  f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
4. 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
5. 0_{+} 0_{+} 2_{+} -1 -6 7 3 0_{+} 1  1^{3} 2_{+} 1  3 3 3  3 3 3
C_{3}  3 3 3 3 3 3
6. 0_{+} 2_{+} 2_{+} -1 -82 64 1^{3} 2_{+} 1  1^{3} 2_{+} 1  3 3 3 3 3 3
C_{3} 3  3  3 3 3 3
7. 0_{+} 2_{+} 2_{+} -1 -82 311 1^{3} 2_{+} 1  1^{3} 2_{+} 1  3 3 3 3 3
C_{3} 3  3 3  3 3  3

Discriminant class: 13

∞ 13 19 a_{1} a_{2} a_{3} Over p = 13 Over p = 19 2 3 5 7 11 17 23 29
G h  f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
8. 0_{+} 1_{+} 0_{+} 1 -3  1^{2} 1_{+} 13 13 2 0_{+} ∗ ∗ 2  2 2 2
C_{2}  2 2 2  2  2