Low Degree Fields Ramifying Above {2, 5, 13}

Discriminant class: −130

∞ 2 5 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  Over p = 13 3 7 11 17 19 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_{+} 1_{+} 0 130  1^{2} 3_{i} −2 −2 1^{2} 1_{+} 5 5 1^{2} 1_{+} 13 13 2 2 2 2 2  2
C_{2} 4  2 2 2 2  2 2

Discriminant class: −65

∞ 2 5 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  Over p = 13  3 7 11 17 19 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_{−} 1_{−} 0 65  1^{2} 2_{i} −1 −1 1^{2} 1_{−} 5∗ 5∗ 1^{2} 1_{−} 13∗ 13∗  2  2
C_{2} 8  2  2 2  2

Discriminant class: −13

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

Discriminant class: −10

∞ 2 5 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  Over p = 13 3 7 11 17 19 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
4. 1_{k} 3_{k} 1_{−} 2_{+} 0 13 -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  2 3 3  2 2

Discriminant class: 13

∞ 2 5 13 a_{1} a_{2} a_{3} Over p = 2 Over p = 5 Over p = 13 3 7 11 17 19 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_{+} 2_{+} 1_{+} 0 -10 -10 1^{3} 2_{+} ∗ 1^{3} 2_{+} ∗ 1^{2} 1_{+} 13 13 3 2 2 2 3 3 2
S_{3}  2 2 3 2 3 2 3 2

Discriminant class: 130

∞ 2 5 13 a_{1} a_{2} a_{3} Over p = 2 Over p = 5 Over p = 13 3 7 11 17 19 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_{+} 3_{+} 1_{+} 1_{+} 0 -130  1^{2} 3_{+} 2 2 1^{2} 1_{+} 5 5 1^{2} 1_{+} 13 13  2  2 2 2
C_{2} 4  2 2  2 2