Basic invariants
| Dimension: | $2$ |
| Group: | $QD_{16}$ |
| Conductor: | \(1475\)\(\medspace = 5^{2} \cdot 59 \) |
| Artin number field: | Galois closure of 8.2.3209046875.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $QD_{16}$ |
| Parity: | odd |
| Projective image: | $D_4$ |
| Projective field: | Galois closure of 4.2.1475.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 + 30\cdot 79 + 15\cdot 79^{2} + 70\cdot 79^{3} + 22\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 14 + 38\cdot 79 + 77\cdot 79^{2} + 11\cdot 79^{3} + 17\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 25 + 78\cdot 79 + 69\cdot 79^{2} + 49\cdot 79^{3} + 9\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 29 + 21\cdot 79 + 53\cdot 79^{2} + 55\cdot 79^{3} + 55\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 51 + 72\cdot 79 + 74\cdot 79^{2} + 14\cdot 79^{3} + 78\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 52 + 60\cdot 79 + 5\cdot 79^{2} + 45\cdot 79^{3} + 46\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 69 + 57\cdot 79^{2} + 12\cdot 79^{3} + 59\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 73 + 13\cdot 79 + 41\cdot 79^{2} + 55\cdot 79^{3} + 26\cdot 79^{4} +O(79^{5})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 8 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 8 }$ | Character values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,4)(2,8)(3,5)(6,7)$ | $-2$ | $-2$ |
| $4$ | $2$ | $(1,4)(2,6)(7,8)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,5,4,3)(2,6,8,7)$ | $0$ | $0$ |
| $4$ | $4$ | $(1,2,4,8)(3,6,5,7)$ | $0$ | $0$ |
| $2$ | $8$ | $(1,6,5,8,4,7,3,2)$ | $-\zeta_{8}^{3} - \zeta_{8}$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,7,5,2,4,6,3,8)$ | $\zeta_{8}^{3} + \zeta_{8}$ | $-\zeta_{8}^{3} - \zeta_{8}$ |