Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(1040\)\(\medspace = 2^{4} \cdot 5 \cdot 13 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.1124864000.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.260.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.1040.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} - 6x^{6} + 14x^{5} + 11x^{4} - 46x^{3} + 44x^{2} - 12x + 1 \)
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The roots of $f$ are computed in $\Q_{ 97 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 24 + 81\cdot 97 + 53\cdot 97^{2} + 12\cdot 97^{3} + 7\cdot 97^{4} + 10\cdot 97^{5} +O(97^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 46 + 67\cdot 97 + 88\cdot 97^{2} + 84\cdot 97^{3} + 63\cdot 97^{4} + 72\cdot 97^{5} +O(97^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 47 + 37\cdot 97 + 21\cdot 97^{2} + 80\cdot 97^{3} + 6\cdot 97^{4} + 59\cdot 97^{5} +O(97^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 58 + 64\cdot 97 + 84\cdot 97^{2} + 32\cdot 97^{3} + 51\cdot 97^{4} + 23\cdot 97^{5} +O(97^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 65 + 31\cdot 97 + 21\cdot 97^{2} + 39\cdot 97^{3} + 95\cdot 97^{4} + 88\cdot 97^{5} +O(97^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 66 + 79\cdot 97 + 84\cdot 97^{2} + 20\cdot 97^{3} + 68\cdot 97^{4} + 61\cdot 97^{5} +O(97^{6})\)
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| $r_{ 7 }$ | $=$ |
\( 85 + 49\cdot 97 + 80\cdot 97^{2} + 35\cdot 97^{3} + 59\cdot 97^{4} + 73\cdot 97^{5} +O(97^{6})\)
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| $r_{ 8 }$ | $=$ |
\( 96 + 72\cdot 97 + 49\cdot 97^{2} + 81\cdot 97^{3} + 35\cdot 97^{4} + 95\cdot 97^{5} +O(97^{6})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,3)(4,6)(7,8)$ | $-2$ |
| $4$ | $2$ | $(1,6)(2,8)(3,7)(4,5)$ | $0$ |
| $4$ | $2$ | $(1,7)(4,6)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,7,5,8)(2,4,3,6)$ | $0$ |
| $2$ | $8$ | $(1,4,8,2,5,6,7,3)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,2,7,4,5,3,8,6)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.