Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(295\)\(\medspace = 5 \cdot 59 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.1514670125.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.295.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.2.1475.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + 5x^{6} - x^{5} + 11x^{4} - 25x^{3} + 14x^{2} - x + 1 \)
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The roots of $f$ are computed in $\Q_{ 311 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 140\cdot 311 + 264\cdot 311^{2} + 292\cdot 311^{3} + 264\cdot 311^{4} +O(311^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 14 + 75\cdot 311 + 126\cdot 311^{2} + 139\cdot 311^{3} + 262\cdot 311^{4} +O(311^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 98 + 148\cdot 311 + 189\cdot 311^{2} + 254\cdot 311^{3} + 25\cdot 311^{4} +O(311^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 100 + 16\cdot 311 + 90\cdot 311^{2} + 77\cdot 311^{3} + 185\cdot 311^{4} +O(311^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 212 + 294\cdot 311 + 220\cdot 311^{2} + 233\cdot 311^{3} + 125\cdot 311^{4} +O(311^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 214 + 162\cdot 311 + 121\cdot 311^{2} + 56\cdot 311^{3} + 285\cdot 311^{4} +O(311^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 298 + 235\cdot 311 + 184\cdot 311^{2} + 171\cdot 311^{3} + 48\cdot 311^{4} +O(311^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 309 + 170\cdot 311 + 46\cdot 311^{2} + 18\cdot 311^{3} + 46\cdot 311^{4} +O(311^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(2,3)(4,5)(6,7)$ | $0$ |
| $4$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ |
| $2$ | $8$ | $(1,3,5,7,8,6,4,2)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,7,4,3,8,2,5,6)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.