Low Degree Fields Ramifying Above {2, 83}

Discriminant class: −166

∞ 2 83 a_{1} a_{2} a_{3} Over p = 2 Over p = 83 3 5 7 11 13 17 19 23
G h  f^{e} c_{w} d s f^{e} c_{w} d s 29 31 37 41 43 47 53 59
1. 1_{k} 3_{−} 1_{k} 0 166  1^{2} 3_{−} 2∗ 2∗ 1^{2} 1_{k} 83 83 2  2
C_{2} 10  2  2  2  2

Discriminant class: −1

∞ 2 83 a_{1} a_{2} a_{3}  Over p = 2  Over p = 83 3 5 7 11 13 17 19 23
G h  f^{e} c_{w} d  s f^{e} c_{w} d s 29 31 37 41 43 47 53 59
2. 1_{k} 2_{i} 2_{+} -1 28 -40 1^{2} 2_{i} −1 −1 1^{3} 2_{+} ∗  2  2 2 3 3 2 2
S_{3} 2  2 3 3 2 2 3 2

Discriminant class: 83

∞ 2 83 a_{1} a_{2} a_{3}  Over p = 2  Over p = 83 3 5 7 11 13 17 19 23
G h  f^{e} c_{w} d  s f^{e} c_{w} d s 29 31 37 41 43 47 53 59
3. 0_{+} 2_{i} 1_{k} 0 -83  1^{2} 2_{i} −∗ −∗ 1^{2} 1_{k} 83 83 2 2 2 2 2  2
C_{2}  2  2 2

Discriminant class: 166

∞ 2 83 a_{1} a_{2} a_{3}  Over p = 2  Over p = 83  3 5 7 11 13 17 19 23
G h  f^{e} c_{w} d  s f^{e} c_{w} d  s 29 31 37 41 43 47 53 59
4. 0_{+} 3_{k} 1_{i} 0 -166  1^{2} 3_{k} −2∗ −2∗ 1^{2} 1_{i} 83∗ 83∗  2  2 2
C_{2}  2 2 2  2