Low Degree Fields Unramified Outside of {∞, 5, 7, 13}

Discriminant class: −455

∞ 5 7 13 a_{1} a_{2} a_{3} Over p = 5 Over p = 7 Over p = 13 2 3 11 17 19 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_{i} 1_{+} 0 455  1^{2} 1_{+} 5 5 1^{2} 1_{i} 7∗ 7∗ 1^{2} 1_{+} 13 13  2  2
C_{2} 20  2 2 2  2

Discriminant class: −91

∞ 5 7 13 a_{1} a_{2} a_{3} Over p = 5  Over p = 7  Over p = 13  2 3 11 17 19 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_{k} 1_{−} 0 91  1^{2} 1_{k} 7 7 1^{2} 1_{−} 13∗ 13∗ 2 2 2 2
C_{2} 2 2 2 2

Discriminant class: −35

∞ 5 7 13 a_{1} a_{2} a_{3} Over p = 5 Over p = 7 Over p = 13 2 3 11 17 19 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_{−} 1_{k} 0_{+} 0 35  1^{2} 1_{−} 5∗ 5∗ 1^{2} 1_{k} 7 7  2  2 2  2
C_{2} 2  2 2 2  2 2 2 2
4. 1_{k} 1_{−} 1_{k} 2_{+} -1 9 -14 1^{2} 1_{−} 5∗ 5∗ 1^{2} 1_{k} 7 7 1^{3} 2_{+} 1  2 3  3 2 2 3 2
S_{3} 3  2 2 2 3 2 2 2 2

Discriminant class: −7

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

Discriminant class: 1

∞ 5 7 13 a_{1} a_{2} a_{3} Over p = 5 Over p = 7 Over p = 13 2 3 11 17 19 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_{+} 0_{+} 2_{+} 0_{+} -1 -2 1 3 0_{+} 1  1^{3} 2_{+} 1  3 3 3 3 3 3  3
C_{3}  3  3 3 3 3 3
8. 0_{+} 0_{+} 0_{+} 2_{+} -1 -4 -1  3 0_{+} 1  1^{3} 2_{+} 1  3 3 3 3 3 3 3
C_{3}  3 3 3  3 3 3
9. 0_{+} 0_{+} 2_{+} 2_{+} -1 -30 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
10. 0_{+} 0_{+} 2_{+} 2_{+} 1 -30 27 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 7 13 a_{1} a_{2} a_{3} Over p = 5  Over p = 7  Over p = 13 2 3 11 17 19 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_{+} 1_{+} 0_{+} 0_{+} 1 -1  1^{2} 1_{+} 5 5 2 0_{+} ∗ ∗ 2 0_{+} ∗ ∗ 2 2  2  2
C_{2}  2  2 2 2  2

Discriminant class: 13

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

Discriminant class: 65

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