Low Degree Fields Ramifying Above {2, 5, 17}

Discriminant class: −170

∞ 2 5 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  Over p = 17  3 7 11 13 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_{k} 1_{+} 1_{−} 0 170  1^{2} 3_{k} −2∗ −2∗ 1^{2} 1_{+} 5 5 1^{2} 1_{−} 17∗ 17∗  2 2  2
C_{2} 12  2 2 2  2
2. 1_{k} 3_{k} 1_{+} 1_{−} -1 -6 10 1^{2} 3_{k} −2∗ −2∗ 1^{2} 1_{+} 5 5 1^{2} 1_{−} 17∗ 17∗ 3 2 2 3 3 2 3 3
S_{3}  2 2 2 3 3 3 3 2

Discriminant class: −85

∞ 2 5 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  Over p = 17  3 7 11 13 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} 1_{−} 1_{−} 0 85  1^{2} 2_{i} −∗ −∗ 1^{2} 1_{−} 5∗ 5∗ 1^{2} 1_{−} 17∗ 17∗ 2 2  2 2 2 2
C_{2} 4  2  2 2 2

Discriminant class: −17

∞ 2 5 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  Over p = 17 3 7 11 13 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} 2_{i} 2_{+} 1_{+} -1 12 -8 1^{2} 2_{i} −1 −1 1^{3} 2_{+} ∗  1^{2} 1_{+} 17 17 3 3  3 2  2 3
S_{3}  2 2 2 2 3 2 2 2

Discriminant class: −10

∞ 2 5 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  Over p = 17 3 7 11 13 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. 1_{k} 3_{k} 1_{−} 2_{+} 0 -17 -34 1^{2} 3_{k} −2∗ −2∗ 1^{2} 1_{−} 5∗ 5∗ 1^{3} 2_{+} ∗  2 3 3 3  3 2 2
S_{3}  2 3 3 3 2 2

Discriminant class: −5

∞ 2 5 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  Over p = 17 3 7 11 13 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. 1_{k} 2_{i} 1_{+} 2_{+} -1 -11 -29 1^{2} 2_{i} −∗ −∗ 1^{2} 1_{+} 5 5 1^{3} 2_{+} ∗  3 3 2 2 2 3  2
S_{3}  2 3  2 2  3

Discriminant class: 5

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

Discriminant class: 170

∞ 2 5 17 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5  Over p = 17  3 7 11 13 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
8. 0_{+} 3_{−} 1_{+} 1_{−} 0 -170  1^{2} 3_{−} 2∗ 2∗ 1^{2} 1_{+} 5 5 1^{2} 1_{−} 17∗ 17∗ 2  2  2
C_{2} 4  2 2  2  2