Low Degree Fields Ramifying Above {2, 7, 19}

Discriminant class: −266

∞ 2 7 19 a_{1} a_{2} a_{3}  Over p = 2  Over p = 7  Over p = 19 3 5 11 13 17 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_{k} 1_{k} 1_{k} 0 266  1^{2} 3_{k} −2∗ −2∗ 1^{2} 1_{k} 7 7 1^{2} 1_{k} 19 19  2 2 2  2
C_{2} 20  2

Discriminant class: −133

∞ 2 7 19 a_{1} a_{2} a_{3}  Over p = 2  Over p = 7  Over p = 19  3 5 11 13 17 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_{i} 1_{k} 1_{i} 0 133  1^{2} 2_{i} −∗ −∗ 1^{2} 1_{k} 7 7 1^{2} 1_{i} 19∗ 19∗ 2 2 2  2 2 2 2
C_{2} 4  2  2  2 2 2

Discriminant class: −19

∞ 2 7 19 a_{1} a_{2} a_{3} Over p = 2  Over p = 7  Over p = 19  3 5 11 13 17 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} 2_{+} 2_{+} 1_{i} -1 -9 -13 1^{3} 2_{+} ∗  1^{3} 2_{+} 1  1^{2} 1_{i} 19∗ 19∗ 2 3  2 3 3 2 2
S_{3} 3  2 2 3  2 2  2
4. 1_{k} 2_{+} 2_{+} 1_{i} -1 -9 29 1^{3} 2_{+} ∗  1^{3} 2_{+} 1  1^{2} 1_{i} 19∗ 19∗ 2  3 2 3 3 2 2
S_{3} 3  2 2  3 2 2 3 2

Discriminant class: −14

∞ 2 7 19 a_{1} a_{2} a_{3} Over p = 2 Over p = 7 Over p = 19 3 5 11 13 17 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} 3_{+} 1_{i} 2_{+} 0 -19 -114 1^{2} 3_{+} 2 2 1^{2} 1_{i} 7∗ 7∗ 1^{3} 2_{+} 1  3 2 3 2 3 2 2
S_{3} 6  2 2 2 2 2 3 3 2

Discriminant class: 266

∞ 2 7 19 a_{1} a_{2} a_{3}  Over p = 2  Over p = 7  Over p = 19  3 5 11 13 17 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_{+} 3_{−} 1_{i} 1_{i} 0 -266  1^{2} 3_{−} 2∗ 2∗ 1^{2} 1_{i} 7∗ 7∗ 1^{2} 1_{i} 19∗ 19∗ 2  2 2 2
C_{2} 2  2 2  2