Quartic Fields Unramified Outside of {∞, 13, 17}

Discriminant class: 1

∞ 13 17 a_{1} a_{2} a_{3}  Over p = 13  Over p = 17 2 3 5 7 11 19 23 29
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
1. 0_{+} 2_{+} 2_{+} 0 -15 0 1^{2} 1_{+} 13 13 1^{2} 1_{+} 17 17 22 22 22 22 22 22 22 22
V_{4} 1 1^{2} 1_{+} 13 13 1^{2} 1_{+} 17 17 22 22 22 22 22 22

Discriminant class: 13

∞ 13 17 a_{1} a_{2} a_{3}  Over p = 13  Over p = 17  2 3 5 7 11 19 23 29
G h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d  s 31 37 41 43 47 53 59 61
2. 2_{−} 3_{−} 0_{+} -1 2 4 1^{4} 3_{−} 13 13 2 0_{+} ∗  ∗ 4  4 4 4 4 22
C_{4}  3  2 0_{+} ∗  ∗ 4 4 4 22 4  4
3. 2_{−} 3_{−} 2_{+} 1 54 -56 1^{4} 3_{−} 13 13 1^{2} 1_{−} 17∗ 17∗ 4 22 4 4 4 4  22
C_{4} 4 263 1^{2} 1_{−} 17∗ 17∗ 4 4 4 22 4 4 22
4. 2_{−} 1_{−} 2_{+} 2 6 5 1^{2} 1_{−} 13∗ 13∗ 1^{2} 1_{+} 17 17 2 22 4 4 4 2 22 22
D_{4} 2 2 0_{+} ∗ ∗ 1^{2} 1_{+} 17 17 4 4 4 22 2 22 2 22
5. 0_{+} 3_{−} 2_{−} -2 -18 19 1^{4} 3_{−} 13 13 2^{2} 2_{−} 1  V 4 22 2 2 2 4 22 22
D_{4} 2 35 2 2 2 4 22 4 22

Discriminant class: 17

∞ 13 17 a_{1} a_{2} a_{3}  Over p = 13  Over p = 17 2 3 5 7 11 19 23 29
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
6. 0_{+} 0_{+} 3_{+} 1 -6 -1  1^{4} 3_{+} 17 17 22 4 4 4 4 22 4 4
C_{4}  1  4 4 4 22  22 22 4
7. 0_{+} 2_{+} 3_{+} -1 -57 1 1^{2} 1_{+} 13 13 1^{4} 3_{+} 17 17  4 4 4 4  4 4
C_{4} 2  664  1^{2} 1_{+} 13 13  4 4 4 22 22 22  4
8. 2_{−} 2_{−} 1_{+} -1 1 -2 2^{2} 2_{−} 1 V 1^{2} 1_{+} 17 17 22 2 4 4 4 22 2 2
D_{4} 4 4 4 4 22 22 22 22 2
9. 2_{−} 2_{−} 3_{+} 0 17 0 2^{2} 2_{−} 1 V 1^{4} 3_{+} 17 17 22 4 2 2 2 22 4 4
D_{4} 8 17 2 2 2 22 22 4

Discriminant class: 221

∞ 13 17 a_{1} a_{2} a_{3}  Over p = 13  Over p = 17  2 3 5 7 11 19 23 29
G h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d  s 31 37 41 43 47 53 59 61
10. 2_{−} 3_{−} 3_{+}  1 28 152 1^{4} 3_{−} 13 13 1^{4} 3_{+} 17 17 4 4 22 22 22 4 4 4
C_{4} 8  1667  22  4 22 4 4
11. 2_{−} 3_{−} 3_{+}  -1 28 290 1^{4} 3_{−} 13 13 1^{4} 3_{+} 17 17 4 4  4 4 4
C_{4} 40  1225  22 22  4 22 4 4
12. 0_{+} 3_{−} 1_{−}  -1 -11 17 1^{4} 3_{−} 13 13 1^{2} 1_{−} 17∗ 17∗ 4 2 22 22 22 4 2 2
D_{4} 3 2 0_{+} ∗ ∗ 22 22 22 4 22 4 2
13. 2_{−} 1_{−} 3_{+}  -2 10 -9 1^{2} 1_{−} 13∗ 13∗ 1^{4} 3_{+} 17 17 2 4 22 22 22 2 4 4
D_{4} 4  16  2 0_{+} ∗  ∗  22 22 22  2  2 4