New Sextic Fields Unramified Outside of {∞, 3, 13}

Discriminant class: −39

∞  3 13 a_{1} a_{2} a_{3}  Over p = 3  Over p = 13  2 5 7 11 17 19 23 29
G  h  a_{4} a_{5} a_{6} f^{e} c_{w} d s f^{e} c_{w} d  s 31 37 41 43 47 53 59 61
1. 3_{i}  9_{i} 1_{−} 0 6 1 1^{6} 9_{i} 3∗  1^{2} 1_{−} 13∗ 13∗ 42 42 32 42 6 32 6 6
C_{3}^{2}.D_{4}  9 3 1  2 0_{+} ∗  ∗ 32 32 42 3 42 6 42 22
2. 1_{k}  7_{k} 3_{−} 0 -6 -1 1^{3} 4_{+} ∗  2^{2} 2_{−} 1  V 42 42 6 42 32 6 32 32
C_{3}^{2}.D_{4}  9 3 -3 1^{3} 3_{k} 3  1^{2} 1_{+} 13 13 6 6 42 33 42 32 42 22

Discriminant class: −3

∞  3 13 a_{1} a_{2} a_{3}  Over p = 3  Over p = 13 2 5 7 11 17 19 23 29
G  h  a_{4} a_{5} a_{6} f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
3. 1_{k}  5_{k} 4_{−} 0 -6 -4 1^{3} 4_{+} ∗  1^{4} 3_{−} 13 13 6 6 42 6 32 42 32 32
C_{3}^{2}.D_{4}  9 12 -9 1^{2} 1_{k} 3 3 1^{2} 1_{+} 13 13 42 42 222 22 6 32 222 22
4. 3_{i}  9_{i} 4_{−} 0 -6 -9 1^{6} 9_{i} 3∗  1^{4} 3_{−} 13 13 32 32 42 32 6 42 6 6
C_{3}^{2}.D_{4} 4  9 27 30  1^{2} 1_{+} 13 13 42 42 2 22 32 6 2 22

Discriminant class: 1

∞  3 13 a_{1} a_{2} a_{3}  Over p = 3  Over p = 13 2 5 7 11 17 19 23 29
G  h  a_{4} a_{5} a_{6} f^{e} c_{w} d s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
5. 2_{−}  4_{+} 4_{−} -1 -3 1 1^{3} 4_{+} 1  1^{4} 3_{−} 13 13 42 42 42 42 22 42 22 33
C_{3}^{2}.C_{4}  6 0 -3 3 0_{+} 1  1^{2} 1_{+} 13 13 42 42 42 22 42 3 42 3
6. 0_{+}  8_{−} 4_{−} 0 -15 -9 1^{3} 5_{i} 3∗  1^{4} 3_{−} 13 13 42 42 42 42 33 42 3 22
C_{3}^{2}.C_{4}  27 9 -9 1^{3} 3_{i} 3∗  1^{2} 1_{+} 13 13 42 42 42 22 42 22 42 33
7. 2_{−}  8_{+} 4_{−} 0 -6 -9 1^{3} 4_{+} 1  1^{4} 3_{−} 13 13 42 42 42 42 22 42 22 3
C_{3}^{2}.C_{4}  3  9 27 17 1^{3} 4_{+} 1  1^{2} 1_{+} 13 13 42 42 42 22 42 33 42 33
8. 0_{+} 10_{−} 4_{−} 0 -18 -9 1^{3} 5_{i} 3∗  1^{4} 3_{−} 13 13 42 42 42 42 3 42 33 22
C_{3}^{2}.C_{4}  81 81 -9 1^{3} 5_{i} 3∗  1^{2} 1_{+} 13 13 42 42 42 22 42 22 42 3
9. 2_{−}  8_{−} 2_{+} 3 3 2 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  5 33 42 42 5 42 42 42
A_{6}  0 0 3 1^{3} 3_{i} 3∗  3 0_{+} 1  5 5 5 5 3 42 5 5
10. 2_{−} 10_{−} 4_{+} 3 3 1 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  5 3 42 42 5 42 42 42
A_{6}  3  0 117 195 1^{3} 5_{i} 3∗  1^{3} 2_{+} 1  5 5 5 5 33 42 5 5