Quartic Fields Unramified Outside of {∞, 13, 41}

Discriminant class: 1

∞ 13 41 a_{1} a_{2} a_{3} Over p = 13  Over p = 41 2 3 5 7 11 17 19 23
G h a_{4} f^{e} c_{w} d s f^{e} c_{w} d s 29 31 37 43 47 53 59 61
1. 0_{+} 2_{−} 2_{−} 0 -27 0 2^{2} 2_{−} 1 V 2^{2} 2_{−} 1 V 22 22 22 22 22 22 22
V_{4} 49 22 22 22 22 22 22

Discriminant class: 13

∞ 13 41 a_{1} a_{2} a_{3}  Over p = 13  Over p = 41 2 3 5 7 11 17 19 23
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d s 29 31 37 43 47 53 59 61
2. 2_{−} 3_{−} 0_{+}  -1 2  4 1^{4} 3_{−} 13 13 4 0_{+} ∗ ∗ 4  4 4 4 22 4 22
C_{4}  3  4 4 22 4  4
3. 2_{−} 3_{−} 2_{+}  1 132 -134 1^{4} 3_{−} 13 13 2^{2} 2_{+} ∗ ∗ 4 22 4 4 4  4 22
C_{4} 10  1433  22 4 4 22 4 22 4
4. 2_{−} 3_{−} 2_{+}  0 -13 -52 1^{4} 3_{−} 13 13 1^{3} 2_{+} ∗  2 3 4 4 4 3 4 3
S_{4} 16  208  3 2 4 3 4 22 2 22

Discriminant class: 41

∞ 13 41 a_{1} a_{2} a_{3} Over p = 13  Over p = 41 2 3 5 7 11 17 19 23
G h a_{4} f^{e} c_{w} d s f^{e} c_{w} d s 29 31 37 43 47 53 59 61
5. 0_{+} 0_{+} 3_{+}  -1 -15 -18 4 0_{+} ∗ ∗ 1^{4} 3_{+} 41 41 22 4 22 4 4 4 4
C_{4}  -4  4  22 4 4  22
6. 0_{+} 2_{+} 3_{+}  -1 -138 -264 2^{2} 2_{+} ∗ ∗ 1^{4} 3_{+} 41 41  4  4 4 4 4
C_{4} 2  1472  4 22 22 22 4 4 22 22

Discriminant class: 533

∞ 13 41 a_{1} a_{2}  a_{3}  Over p = 13  Over p = 41  2 3 5 7 11 17 19 23
G h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d  s 29 31 37 43 47 53 59 61
7. 2_{−} 3_{+} 3_{−}  -1 67 433 1^{4} 3_{+} 13∗ 13∗ 1^{4} 3_{−} 41∗ 41∗ 4 4 4 22 22 4 22 22
C_{4} 20  9467  4 4 4  4 4 22
8. 2_{−} 3_{+} 3_{−}  1 67 -1499 1^{4} 3_{+} 13∗ 13∗ 1^{4} 3_{−} 41∗ 41∗ 4 4 4  4  22
C_{4} 68  5203  4 4 4  22 4 4 22