Low Degree Fields Ramifying Above {2, 5, 19}

Discriminant class: −190

∞ 2 5 19 a_{1} a_{2} a_{3} Over p = 2 Over p = 5 Over p = 19 3 7 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_{+} 1_{−} 1_{k} 0 190  1^{2} 3_{+} 2 2 1^{2} 1_{−} 5∗ 5∗ 1^{2} 1_{k} 19 19 2 2 2 2 2 2  2
C_{2} 4  2 2  2 2  2 2

Discriminant class: −10

∞ 2 5 19 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  Over p = 19 3 7 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} 3_{k} 1_{−} 2_{+} -1 -6 26 1^{2} 3_{k} −2∗ −2∗ 1^{2} 1_{−} 5∗ 5∗ 1^{3} 2_{+} 1  2 3 3  2 3 2 2
S_{3} 3  3 3 2  3  2 2

Discriminant class: −2

∞ 2 5 19 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  Over p = 19 3 7 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} 3_{i} 2_{+} 2_{+} -1 32 292 1^{2} 3_{i} −2 −2 1^{3} 2_{+} ∗  1^{3} 2_{+} 1  2 3 2 3 2 2 2
S_{3} 3  2 3  2 2 3 2 3
4. 1_{k} 3_{i} 2_{+} 2_{+} -1 32 -88 1^{2} 3_{i} −2 −2 1^{3} 2_{+} ∗  1^{3} 2_{+} 1  3 2  2 3 2 2 2
S_{3} 3  2  2 2 3 2

Discriminant class: 5

∞ 2 5 19 a_{1} a_{2} a_{3} Over p = 2 Over p = 5 Over p = 19 3 7 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. 0_{+} 2_{+} 1_{+} 2_{+} -1 -25 45 1^{3} 2_{+} ∗  1^{2} 1_{+} 5 5 1^{3} 2_{+} 1  2 2 3 2 2 2 3 3
S_{3} 3  2  2 2 2 3 3 2

Discriminant class: 95

∞ 2 5 19 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  Over p = 19 3 7 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_{+} 2_{i} 1_{+} 1_{k} 0 -95  1^{2} 2_{i} −1 −1 1^{2} 1_{+} 5 5 1^{2} 1_{k} 19 19 2  2  2  2
C_{2} 2  2  2

Discriminant class: 190

∞ 2 5 19 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  Over p = 19  3 7 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
7. 0_{+} 3_{i} 1_{−} 1_{i} 0 -190  1^{2} 3_{i} −2 −2 1^{2} 1_{−} 5∗ 5∗ 1^{2} 1_{i} 19∗ 19∗  2 2
C_{2} 2  2 2 2  2 2 2