Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(1156\)\(\medspace = 2^{2} \cdot 17^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.26261675072.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.4.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.2.19652.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} + 10x^{6} - 11x^{5} + 32x^{4} - 61x^{3} + 41x^{2} - 132x + 18 \)
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The roots of $f$ are computed in $\Q_{ 293 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 60 + 230\cdot 293 + 253\cdot 293^{2} + 86\cdot 293^{3} + 142\cdot 293^{4} +O(293^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 64 + 42\cdot 293 + 81\cdot 293^{2} + 37\cdot 293^{3} + 157\cdot 293^{4} +O(293^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 123 + 245\cdot 293 + 92\cdot 293^{2} + 97\cdot 293^{3} + 240\cdot 293^{4} +O(293^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 144 + 199\cdot 293 + 177\cdot 293^{2} + 23\cdot 293^{3} + 166\cdot 293^{4} +O(293^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 148 + 174\cdot 293 + 14\cdot 293^{2} + 199\cdot 293^{3} + 260\cdot 293^{4} +O(293^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 157 + 279\cdot 293 + 228\cdot 293^{2} + 200\cdot 293^{3} + 73\cdot 293^{4} +O(293^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 187 + 158\cdot 293 + 151\cdot 293^{2} + 113\cdot 293^{3} + 274\cdot 293^{4} +O(293^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 290 + 134\cdot 293 + 171\cdot 293^{2} + 120\cdot 293^{3} + 150\cdot 293^{4} +O(293^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,7)(3,4)(6,8)$ | $-2$ |
| $4$ | $2$ | $(2,4)(3,7)(6,8)$ | $0$ |
| $4$ | $2$ | $(1,7)(2,5)(3,6)(4,8)$ | $0$ |
| $2$ | $4$ | $(1,6,5,8)(2,3,7,4)$ | $0$ |
| $2$ | $8$ | $(1,7,6,4,5,2,8,3)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,4,8,7,5,3,6,2)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.