Low Degree Fields Ramifying Above {2, 13, 23}

Discriminant class: −598

∞ 2 13 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 13  Over p = 23  3 5 7 11 17 19 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_{i} 0 598  1^{2} 3_{−} 2∗ 2∗ 1^{2} 1_{−} 13∗ 13∗ 1^{2} 1_{i} 23∗ 23∗ 2 2  2 2 2 2 2
C_{2} 8  2 2  2  2

Discriminant class: −299

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

Discriminant class: −13

∞ 2 13 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 13  Over p = 23 3 5 7 11 17 19 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_{i} 1_{+} 2_{+} -1 8 -98 1^{2} 2_{i} −∗ −∗ 1^{2} 1_{+} 13 13 1^{3} 2_{+} ∗  2 2  3 3 3 3 3
S_{3}  2 2 2 3 3 3  3

Discriminant class: −1

∞ 2 13 23 a_{1} a_{2}  a_{3}  Over p = 2  Over p = 13 Over p = 23 3 5 7 11 17 19 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} 2_{i} 2_{+} 2_{+} -1 100 -742 1^{2} 2_{i} −1 −1 1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2 3 2 2 3 2  2
S_{3} 3  3 3 2 2  2  2
5. 1_{k} 2_{i} 2_{+} 2_{+} 1 100 -1052 1^{2} 2_{i} −1 −1 1^{3} 2_{+} 1  1^{3} 2_{+} ∗  2  2 2 3 2 3 2
S_{3} 6  3  2 2 3 2 3 2

Discriminant class: 299

∞ 2 13 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 13  Over p = 23 3 5 7 11 17 19 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 -299  1^{2} 2_{i} −∗ −∗ 1^{2} 1_{+} 13 13 1^{2} 1_{k} 23 23 2  2 2 2 2
C_{2} 2  2  2  2 2

Discriminant class: 598

∞ 2 13 23 a_{1} a_{2} a_{3}  Over p = 2  Over p = 13  Over p = 23 3 5 7 11 17 19 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_{−} 1_{k} 0 -598  1^{2} 3_{k} −2∗ −2∗ 1^{2} 1_{−} 13∗ 13∗ 1^{2} 1_{k} 23 23  2 2  2  2
C_{2} 2  2 2 2  2