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 8.0.23973872753.1 |
| 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
| $f(x)$ | $=$ |
\( x^{8} - x^{7} + 8x^{6} - 9x^{5} + 29x^{4} - 23x^{3} + 50x^{2} - 25x + 51 \)
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The roots of $f$ are computed in $\Q_{ 491 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 8 + 183\cdot 491 + 370\cdot 491^{2} + 93\cdot 491^{3} + 157\cdot 491^{4} +O(491^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 80 + 263\cdot 491 + 248\cdot 491^{2} + 271\cdot 491^{3} + 467\cdot 491^{4} +O(491^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 90 + 452\cdot 491 + 257\cdot 491^{2} + 23\cdot 491^{3} + 11\cdot 491^{4} +O(491^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 118 + 218\cdot 491 + 128\cdot 491^{2} + 430\cdot 491^{3} + 488\cdot 491^{4} +O(491^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 138 + 398\cdot 491 + 234\cdot 491^{2} + 413\cdot 491^{3} + 274\cdot 491^{4} +O(491^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 309 + 23\cdot 491 + 451\cdot 491^{2} + 297\cdot 491^{3} + 58\cdot 491^{4} +O(491^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 336 + 157\cdot 491 + 304\cdot 491^{2} + 274\cdot 491^{3} + 81\cdot 491^{4} +O(491^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 395 + 267\cdot 491 + 459\cdot 491^{2} + 158\cdot 491^{3} + 424\cdot 491^{4} +O(491^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,5)(2,8)(3,6)(4,7)$ | $-2$ | |
| $4$ | $2$ | $(1,4)(3,6)(5,7)$ | $0$ | |
| $4$ | $2$ | $(1,8)(2,5)(3,7)(4,6)$ | $0$ | ✓ |
| $2$ | $4$ | $(1,7,5,4)(2,3,8,6)$ | $0$ | |
| $2$ | $8$ | $(1,6,7,2,5,3,4,8)$ | $-\zeta_{8}^{3} + \zeta_{8}$ | |
| $2$ | $8$ | $(1,2,4,6,5,8,7,3)$ | $\zeta_{8}^{3} - \zeta_{8}$ |