Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(111\)\(\medspace = 3 \cdot 37 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.50602347.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.111.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.333.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} + x^{6} - 6x^{5} + 9x^{4} - 6x^{3} + x^{2} - x + 1 \)
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The roots of $f$ are computed in $\Q_{ 127 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 22 + 21\cdot 127 + 116\cdot 127^{2} + 9\cdot 127^{3} + 72\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 26 + 122\cdot 127 + 27\cdot 127^{2} + 20\cdot 127^{3} + 10\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 44 + 13\cdot 127 + 43\cdot 127^{2} + 33\cdot 127^{3} + 40\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 47 + 79\cdot 127 + 111\cdot 127^{2} + 84\cdot 127^{3} + 123\cdot 127^{4} +O(127^{5})\)
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| $r_{ 5 }$ | $=$ |
\( 52 + 25\cdot 127 + 117\cdot 127^{2} + 61\cdot 127^{3} + 2\cdot 127^{4} +O(127^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 95 + 37\cdot 127 + 101\cdot 127^{2} + 115\cdot 127^{3} + 67\cdot 127^{4} +O(127^{5})\)
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| $r_{ 7 }$ | $=$ |
\( 100 + 50\cdot 127 + 105\cdot 127^{2} + 107\cdot 127^{3} + 92\cdot 127^{4} +O(127^{5})\)
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| $r_{ 8 }$ | $=$ |
\( 123 + 30\cdot 127 + 12\cdot 127^{2} + 74\cdot 127^{3} + 98\cdot 127^{4} +O(127^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,3)(4,7)(6,8)$ | $-2$ |
| $4$ | $2$ | $(1,8)(2,3)(5,6)$ | $0$ |
| $4$ | $2$ | $(1,3)(2,5)(4,8)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,8,5,6)(2,4,3,7)$ | $0$ |
| $2$ | $8$ | $(1,4,8,3,5,7,6,2)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,3,6,4,5,2,8,7)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.