Basic invariants
| Dimension: | $2$ |
| Group: | $QD_{16}$ |
| Conductor: | \(2475\)\(\medspace = 3^{2} \cdot 5^{2} \cdot 11 \) |
| Artin stem field: | Galois closure of 8.2.15160921875.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $QD_{16}$ |
| Parity: | odd |
| Determinant: | 1.11.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.2.2475.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} - 4x^{6} - 16x^{5} - 14x^{4} - 5x^{3} + 5x^{2} + 10x - 5 \)
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The roots of $f$ are computed in $\Q_{ 179 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 18 + 58\cdot 179 + 2\cdot 179^{2} + 27\cdot 179^{3} + 106\cdot 179^{4} +O(179^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 57 + 61\cdot 179 + 125\cdot 179^{2} + 172\cdot 179^{3} + 28\cdot 179^{4} +O(179^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 104 + 158\cdot 179 + 30\cdot 179^{2} + 89\cdot 179^{3} + 124\cdot 179^{4} +O(179^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 110 + 172\cdot 179 + 91\cdot 179^{2} + 12\cdot 179^{3} + 42\cdot 179^{4} +O(179^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 126 + 108\cdot 179 + 135\cdot 179^{2} + 22\cdot 179^{3} + 33\cdot 179^{4} +O(179^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 140 + 22\cdot 179 + 156\cdot 179^{2} + 165\cdot 179^{3} + 102\cdot 179^{4} +O(179^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 168 + 105\cdot 179 + 143\cdot 179^{2} + 67\cdot 179^{3} + 62\cdot 179^{4} +O(179^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 173 + 27\cdot 179 + 30\cdot 179^{2} + 158\cdot 179^{3} + 36\cdot 179^{4} +O(179^{5})\)
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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 value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,4)(3,8)(5,6)$ | $-2$ |
| $4$ | $2$ | $(1,7)(2,5)(4,6)$ | $0$ |
| $2$ | $4$ | $(1,3,7,8)(2,5,4,6)$ | $0$ |
| $4$ | $4$ | $(1,5,7,6)(2,8,4,3)$ | $0$ |
| $2$ | $8$ | $(1,5,3,4,7,6,8,2)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,6,3,2,7,5,8,4)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.