Low Degree Fields Ramifying Above {2, 19}

Discriminant class: −38

∞ 2 19 a_{1} a_{2} a_{3} Over p = 2 Over p = 19 3 5 7 11 13 17 23 29
G h  f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
1. 1_{k} 3_{−} 1_{k} 0 38  1^{2} 3_{−} 2∗ 2∗ 1^{2} 1_{k} 19 19  2  2
C_{2} 6  2  2 2  2
2. 1_{k} 3_{−} 1_{k} -1 -2 -2 1^{2} 3_{−} 2∗ 2∗ 1^{2} 1_{k} 19 19 3 2 3 2 3 3 3 3
S_{3} 2 2 2 3 3 2

Discriminant class: −19

∞ 2 19 a_{1} a_{2} a_{3} Over p = 2  Over p = 19  3 5 7 11 13 17 23 29
G h  f^{e} c_{w} d s f^{e} c_{w} d  s 31 37 41 43 47 53 59 61
3. 1_{k} 2_{+} 1_{i} 0 -2 -2 1^{3} 2_{+} ∗  1^{2} 1_{i} 19∗ 19∗ 2 3 3 3 2 3  2
S_{3}  2 2 2 3 3 2 2 3

Discriminant class: −2

∞ 2 19 a_{1} a_{2} a_{3}  Over p = 2  Over p = 19 3 5 7 11 13 17 23 29
G h  f^{e} c_{w} d  s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
4. 1_{k} 3_{i} 2_{+} -1 13 7 1^{2} 3_{i} −2 −2 1^{3} 2_{+} 1  3 2 2 3 2  2 2
S_{3} 3  2 2 3  2 2 3 2

Discriminant class: 19

∞ 2 19 a_{1} a_{2} a_{3}  Over p = 2  Over p = 19 3 5 7 11 13 17 23 29
G h  f^{e} c_{w} d  s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
5. 0_{+} 2_{i} 1_{k} 0 -19  1^{2} 2_{i} −∗ −∗ 1^{2} 1_{k} 19 19  2 2 2  2 2
C_{2}  2 2 2 2 2

Discriminant class: 38

∞ 2 19 a_{1} a_{2} a_{3}  Over p = 2  Over p = 19  3 5 7 11 13 17 23 29
G h  f^{e} c_{w} d  s f^{e} c_{w} d  s 31 37 41 43 47 53 59 61
6. 0_{+} 3_{k} 1_{i} 0 -38  1^{2} 3_{k} −2∗ −2∗ 1^{2} 1_{i} 19∗ 19∗ 2 2 2  2
C_{2}  2  2  2 2