Low Degree Fields Unramified Outside of {∞, 13, 17, 19}

Discriminant class: −4199

∞ 13 17 19 a_{1} a_{2} a_{3}  Over p = 13  Over p = 17  Over p = 19 2 3 5 7 11 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 4199  1^{2} 1_{−} 13∗ 13∗ 1^{2} 1_{+} 17 17 1^{2} 1_{k} 19 19  2  2
C_{2} 88  2 2  2 2  2

Discriminant class: −323

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

Discriminant class: −247

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

Discriminant class: −19

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

∞ 13 17 19 a_{1} a_{2} a_{3} Over p = 13 Over p = 17 Over p = 19 2 3 5 7 11 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. 0_{+} 2_{+} 0_{+} 0_{+} -1 -4 -1 1^{3} 2_{+} 1  3 0_{+} 1  3 0_{+} 1  3 3  3 3 3 3
C_{3}  3 3 3  3 3 3
7. 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
8. 0_{+} 2_{+} 0_{+} 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
9. 0_{+} 2_{+} 0_{+} 2_{+} -1 -82 64 1^{3} 2_{+} 1  3 0_{+} 1  1^{3} 2_{+} 1  3 3 3 3 3
C_{3} 3  3  3 3 3 3  3

Discriminant class: 13

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

Discriminant class: 17

∞ 13 17 19 a_{1} a_{2} a_{3} Over p = 13  Over p = 17  Over p = 19 2 3 5 7 11 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_{+} 1_{+} 0_{+} 1 -4  1^{2} 1_{+} 17 17  2 2 2 2 2 2 2
C_{2}  2 2  2
12. 0_{+} 2_{+} 1_{+} 2_{+} -1 -82 -183 1^{3} 2_{+} 1  1^{2} 1_{+} 17 17 1^{3} 2_{+} 1  3 2 2 2 2 2 2 2
S_{3} 9  2 2 3 3  3 2

Discriminant class: 221

∞ 13 17 19 a_{1} a_{2} a_{3}  Over p = 13  Over p = 17  Over p = 19 2 3 5 7 11 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_{+} 1_{+} 1_{+} 0_{+} 0 -221  1^{2} 1_{+} 13 13 1^{2} 1_{+} 17 17 2 0_{+} ∗ ∗ 2 2  2 2
C_{2} 2  2  2 2 2