Low Degree Fields Unramified Outside of {∞, 7, 17, 19}

Discriminant class: −323

∞ 7 17 19 a_{1} a_{2} a_{3} Over p = 7  Over p = 17  Over p = 19  2 3 5 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} 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: −119

∞ 7 17 19 a_{1} a_{2} a_{3} Over p = 7  Over p = 17  Over p = 19 2 3 5 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} 1_{k} 1_{−} 0_{+} 0 119  1^{2} 1_{k} 7 7 1^{2} 1_{−} 17∗ 17∗ 2 0_{+} ∗ ∗  2 2 2 2
C_{2} 10  2  2  2

Discriminant class: −19

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

Discriminant class: −7

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

Discriminant class: 1

∞ 7 17 19 a_{1} a_{2} a_{3} Over p = 7  Over p = 17 Over p = 19 2 3 5 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
7. 0_{+} 2_{+} 0_{+} 0_{+} -1 -2 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
8. 0_{+} 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 3
9. 0_{+} 2_{+} 0_{+} 2_{+} -1 -44 -69 1^{3} 2_{+} 1  1^{3} 2_{+} 1  3  3 3 3  3 3
C_{3} 3  3 3 3  3  3
10. 0_{+} 2_{+} 0_{+} 2_{+} 1 -44 -64 1^{3} 2_{+} 1  3 0_{+} 1  1^{3} 2_{+} 1  3  3 3 3 3 3
C_{3} 3  3 3 3 3  3 3

Discriminant class: 17

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

Discriminant class: 133

∞ 7 17 19 a_{1} a_{2} a_{3}  Over p = 7  Over p = 17  Over p = 19 2 3 5 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
12. 0_{+} 1_{i} 0_{+} 1_{k} 0 -133  1^{2} 1_{i} 7∗ 7∗ 2 0_{+} ∗ ∗ 1^{2} 1_{k} 19 19 2  2  2
C_{2}  2  2 2  2 2

Discriminant class: 2261

∞ 7 17 19 a_{1}  a_{2} a_{3} Over p = 7  Over p = 17  Over p = 19 2 3 5 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. 0_{+} 1_{k} 1_{−} 1_{k} 0 -2261  1^{2} 1_{k} 7 7 1^{2} 1_{−} 17∗ 17∗ 1^{2} 1_{k} 19 19 2 2  2  2  2
C_{2} 2  2  2 2  2