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 8.2.128361875.1 |
| 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
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} + 7x^{6} - 10x^{5} + 7x^{4} + x^{3} - 9x^{2} + 8x - 1 \)
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The roots of $f$ are computed in $\Q_{ 311 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 31 + 58\cdot 311 + 147\cdot 311^{2} + 199\cdot 311^{3} + 12\cdot 311^{4} +O(311^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 67 + 215\cdot 311 + 199\cdot 311^{2} + 159\cdot 311^{3} + 9\cdot 311^{4} +O(311^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 117 + 310\cdot 311 + 110\cdot 311^{2} + 137\cdot 311^{3} + 246\cdot 311^{4} +O(311^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 161 + 33\cdot 311 + 154\cdot 311^{2} + 180\cdot 311^{3} + 156\cdot 311^{4} +O(311^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 170 + 290\cdot 311 + 105\cdot 311^{2} + 43\cdot 311^{3} + 186\cdot 311^{4} +O(311^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 181 + 225\cdot 311 + 278\cdot 311^{2} + 200\cdot 311^{3} + 183\cdot 311^{4} +O(311^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 256 + 41\cdot 311 + 79\cdot 311^{2} + 142\cdot 311^{3} + 228\cdot 311^{4} +O(311^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 264 + 68\cdot 311 + 168\cdot 311^{2} + 180\cdot 311^{3} + 220\cdot 311^{4} +O(311^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,8)(4,7)(5,6)$ | $-2$ |
| $4$ | $2$ | $(1,5)(2,4)(3,6)(7,8)$ | $0$ |
| $4$ | $2$ | $(1,4)(3,7)(5,6)$ | $0$ |
| $2$ | $4$ | $(1,7,3,4)(2,6,8,5)$ | $0$ |
| $2$ | $8$ | $(1,6,7,8,3,5,4,2)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,8,4,6,3,2,7,5)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.