Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(799\)\(\medspace = 17 \cdot 47 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.8671400783.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.799.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.2.13583.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} + 2x^{6} + x^{5} - 15x^{4} + 23x^{3} + 6x^{2} - 27x + 19 \)
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The roots of $f$ are computed in $\Q_{ 491 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 65 + 330\cdot 491 + 432\cdot 491^{2} + 148\cdot 491^{3} + 252\cdot 491^{4} +O(491^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 134 + 171\cdot 491 + 138\cdot 491^{2} + 15\cdot 491^{3} + 241\cdot 491^{4} +O(491^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 230 + 433\cdot 491 + 74\cdot 491^{2} + 144\cdot 491^{3} + 429\cdot 491^{4} +O(491^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 247 + 218\cdot 491 + 154\cdot 491^{2} + 442\cdot 491^{3} + 446\cdot 491^{4} +O(491^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 251 + 349\cdot 491 + 462\cdot 491^{2} + 138\cdot 491^{3} + 36\cdot 491^{4} +O(491^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 288 + 393\cdot 491 + 95\cdot 491^{2} + 381\cdot 491^{3} + 103\cdot 491^{4} +O(491^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 319 + 473\cdot 491 + 396\cdot 491^{2} + 275\cdot 491^{3} + 441\cdot 491^{4} +O(491^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 431 + 84\cdot 491 + 208\cdot 491^{2} + 417\cdot 491^{3} + 12\cdot 491^{4} +O(491^{5})\)
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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 | Complex conjugation |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,2)(3,8)(4,5)(6,7)$ | $-2$ | |
| $4$ | $2$ | $(1,3)(2,8)(4,6)(5,7)$ | $0$ | |
| $4$ | $2$ | $(3,6)(4,5)(7,8)$ | $0$ | ✓ |
| $2$ | $4$ | $(1,4,2,5)(3,7,8,6)$ | $0$ | |
| $2$ | $8$ | $(1,6,5,8,2,7,4,3)$ | $\zeta_{8}^{3} - \zeta_{8}$ | |
| $2$ | $8$ | $(1,8,4,6,2,3,5,7)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |