Low Degree Fields Ramifying Above {2, 13}

Discriminant class: −26

∞ 2 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 13  3 5 7 11 17 19 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_{k} 1_{−} 0 26  1^{2} 3_{k} −2∗ −2∗ 1^{2} 1_{−} 13∗ 13∗  2  2 2 2
C_{2} 6  2  2 2 2
2. 1_{k} 3_{k} 1_{−} 0 -1 -2 1^{2} 3_{k} −2∗ −2∗ 1^{2} 1_{−} 13∗ 13∗ 3 3 3 2 3 2 2 2
S_{3}  3 2 3 3 2 2 2

Discriminant class: −13

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

Discriminant class: −1

∞ 2 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 13 3 5 7 11 17 19 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} 2_{i} 2_{+} -1 -4 12 1^{2} 2_{i} −1 −1 1^{3} 2_{+} 1  2 3 2 2 3 2 2 3
S_{3} 3  2 3 3 2 2 3 2 3

Discriminant class: 26

∞ 2 13 a_{1} a_{2} a_{3}  Over p = 2  Over p = 13  3 5 7 11 17 19 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_{+} 3_{−} 1_{−} 0 -26  1^{2} 3_{−} 2∗ 2∗ 1^{2} 1_{−} 13∗ 13∗ 2  2  2
C_{2} 2  2  2 2 2 2  2