Low Degree Fields Ramifying Above {2, 11, 19}

Discriminant class: −418

∞ 2 11 19 a_{1} a_{2} a_{3}  Over p = 2  Over p = 11  Over p = 19 3 5 7 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_{i} 1_{i} 1_{k} 0 418  1^{2} 3_{i} −2 −2 1^{2} 1_{i} 11∗ 11∗ 1^{2} 1_{k} 19 19 2 2  2 2 2 2
C_{2} 8  2 2  2  2

Discriminant class: −209

∞ 2 11 19 a_{1} a_{2} a_{3}  Over p = 2  Over p = 11  Over p = 19  3 5 7 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} 0 209  1^{2} 2_{i} −1 −1 1^{2} 1_{k} 11 11 1^{2} 1_{i} 19∗ 19∗  2 2
C_{2} 20  2  2 2  2

Discriminant class: −38

∞ 2 11 19 a_{1} a_{2} a_{3} Over p = 2 Over p = 11 Over p = 19 3 5 7 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_{−} 2_{+} 1_{k} 0 11 -22 1^{2} 3_{−} 2∗ 2∗ 1^{3} 2_{+} ∗ 1^{2} 1_{k} 19 19 3 2 3 3 3 2
S_{3} 6  3 2 2 3 3 3 2
4. 1_{k} 3_{−} 2_{+} 1_{k} -1 -18 46 1^{2} 3_{−} 2∗ 2∗ 1^{3} 2_{+} ∗  1^{2} 1_{k} 19 19 3 2  3 3  3 2
S_{3} 3  3 2 2 3 3  2 3
5. 1_{k} 3_{−} 2_{+} 1_{k} -1 -18 90 1^{2} 3_{−} 2∗ 2∗ 1^{3} 2_{+} ∗  1^{2} 1_{k} 19 19  2 3 3 3 3  2
S_{3} 3  3 2 2 3  3 2 3

Discriminant class: −22

∞ 2 11 19 a_{1} a_{2} a_{3} Over p = 2 Over p = 11 Over p = 19 3 5 7 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. 1_{k} 3_{−} 1_{k} 2_{+} -1 32 -12 1^{2} 3_{−} 2∗ 2∗ 1^{2} 1_{k} 11 11 1^{3} 2_{+} 1 2 2 2 3 2 3
S_{3} 3  2 2 3 3 2 2 3 2

Discriminant class: 11

∞ 2 11 19 a_{1} a_{2} a_{3}  Over p = 2  Over p = 11  Over p = 19 3 5 7 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_{+} 2_{i} 1_{k} 2_{+} 0 -38 -76 1^{2} 2_{i} −∗ −∗ 1^{2} 1_{k} 11 11 1^{3} 2_{+} 1  2 3 3 2 2 2 2 2
S_{3} 3  3 2  2 3 2 2 2

Discriminant class: 418

∞ 2 11 19 a_{1} a_{2} a_{3} Over p = 2  Over p = 11  Over p = 19  3 5 7 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
8. 0_{+} 3_{+} 1_{k} 1_{i} 0 -418  1^{2} 3_{+} 2 2 1^{2} 1_{k} 11 11 1^{2} 1_{i} 19∗ 19∗  2 2 2 2  2 2
C_{2} 2