Low Degree Fields Ramifying Above {2, 19, 23}

Discriminant class: −874

∞ 2 19 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 19  Over p = 23 3 5 7 11 13 17 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} 0 874  1^{2} 3_{k} −2∗ −2∗ 1^{2} 1_{k} 19 19 1^{2} 1_{k} 23 23 2  2  2
C_{2} 20  2 2 2 2 2 2

Discriminant class: −437

∞ 2 19 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 19  Over p = 23 3 5 7 11 13 17 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_{i} 1_{k} 0 437  1^{2} 2_{i} −∗ −∗ 1^{2} 1_{i} 19∗ 19∗ 1^{2} 1_{k} 23 23  2  2 2 2
C_{2} 20  2  2  2 2

Discriminant class: −46

∞ 2 19 23 a_{1} a_{2} a_{3} Over p = 2  Over p = 19  Over p = 23  3 5 7 11 13 17 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_{i} 0 19 -38 1^{2} 3_{+} 2 2 1^{3} 2_{+} 1  1^{2} 1_{i} 23∗ 23∗ 2 3 2 3 2 2 2 3
S_{3} 6 3 3 3 3 2

Discriminant class: −2

∞ 2 19 23 a_{1} a_{2}  a_{3}  Over p = 2  Over p = 19 Over p = 23 3 5 7 11 13 17 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} 3_{i} 2_{+} 2_{+} -1 146 178 1^{2} 3_{i} −2 −2 1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 2 3 2 3 2 2
S_{3} 3  2 3 3 2 2 3 2 3
5. 1_{k} 3_{i} 2_{+} 2_{+} -1 146 -1570 1^{2} 3_{i} −2 −2 1^{3} 2_{+} 1  1^{3} 2_{+} ∗  3 2 2 3 2 3 2 2
S_{3} 3  2 3 3 2 2  2

Discriminant class: 19

∞ 2 19 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 19  Over p = 23 3 5 7 11 13 17 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_{k} 2_{+} 0 -46 -92 1^{2} 2_{i} −∗ −∗ 1^{2} 1_{k} 19 19 1^{3} 2_{+} ∗  3 3 2 2 2 3 2 3
S_{3} 2 2 2 2 2 3 3

Discriminant class: 38

∞ 2 19 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 19  Over p = 23 3 5 7 11 13 17 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_{i} 2_{+} -1 -38 86 1^{2} 3_{k} −2∗ −2∗ 1^{2} 1_{i} 19∗ 19∗ 1^{3} 2_{+} ∗  2 2 2 3 3 3 3
S_{3}  3 2  2 3 2 2 2

Discriminant class: 874

∞ 2 19 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 19  Over p = 23  3 5 7 11 13 17 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_{i} 1_{i} 0 -874  1^{2} 3_{−} 2∗ 2∗ 1^{2} 1_{i} 19∗ 19∗ 1^{2} 1_{i} 23∗ 23∗  2  2  2
C_{2} 6  2 2  2  2
9. 0_{+} 3_{−} 1_{i} 1_{i} 0 -13 -14 1^{2} 3_{−} 2∗ 2∗ 1^{2} 1_{i} 19∗ 19∗ 1^{2} 1_{i} 23∗ 23∗ 3 3 2 3  2 3 2
S_{3}  2 2 3 3 2 3  2