Low Degree Fields Ramifying Above {2, 5, 7}

Discriminant class: −70

∞ 2 5 7 a_{1} a_{2} a_{3} Over p = 2 Over p = 5 Over p = 7 3 11 13 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} 3_{−} 1_{+} 1_{k} 0 70 1^{2} 3_{−} 2∗ 2∗ 1^{2} 1_{+} 5 5 1^{2} 1_{k} 7 7 2 2 2 2 2 2
C_{2} 4  2

Discriminant class: −35

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

Discriminant class: −10

∞ 2 5 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  Over p = 7 3 11 13 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} 3_{k} 1_{−} 2_{+} -1 5 15 1^{2} 3_{k} −2∗ −2∗ 1^{2} 1_{−} 5∗ 5∗ 1^{3} 2_{+} 1  2 3 3 2 3 3 2 2
S_{3} 3 3 3 2 3 3 2 2

Discriminant class: −5

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

Discriminant class: 2

∞ 2 5 7 a_{1} a_{2} a_{3} Over p = 2 Over p = 5 Over p = 7 3 11 13 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. 0_{+} 3_{+} 2_{+} 2_{+} -1 -23 -13 1^{2} 3_{+} 2 2 1^{3} 2_{+} ∗ 1^{3} 2_{+} 1 2 2 2 3 2 3 2 3
S_{3} 3  2 3 2 3 2 2 2 2

Discriminant class: 35

∞ 2 5 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  Over p = 7  3 11 13 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
6. 0_{+} 2_{i} 1_{−} 1_{i} 0 -35  1^{2} 2_{i} −∗ −∗ 1^{2} 1_{−} 5∗ 5∗ 1^{2} 1_{i} 7∗ 7∗ 2 2
C_{2} 2  2 2  2 2  2

Discriminant class: 70

∞ 2 5 7 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  Over p = 7  3 11 13 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_{+} 3_{k} 1_{+} 1_{i} 0 -70  1^{2} 3_{k} −2∗ −2∗ 1^{2} 1_{+} 5 5 1^{2} 1_{i} 7∗ 7∗  2  2  2
C_{2} 2  2 2 2  2  2