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 8.0.4102893.1 |
| 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
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} + 3x^{6} - 3x^{5} + 5x^{4} - 6x^{3} + 6x^{2} - 3x + 1 \)
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The roots of $f$ are computed in $\Q_{ 127 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 32 + 15\cdot 127 + 60\cdot 127^{2} + 78\cdot 127^{3} + 55\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 51 + 34\cdot 127 + 51\cdot 127^{2} + 96\cdot 127^{3} + 108\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 64 + 34\cdot 127 + 104\cdot 127^{2} + 8\cdot 127^{3} + 65\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 89 + 3\cdot 127 + 48\cdot 127^{2} + 60\cdot 127^{3} + 53\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 93 + 72\cdot 127 + 58\cdot 127^{2} + 57\cdot 127^{3} + 105\cdot 127^{4} +O(127^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 94 + 22\cdot 127 + 57\cdot 127^{2} + 23\cdot 127^{3} + 60\cdot 127^{4} +O(127^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 104 + 113\cdot 127 + 85\cdot 127^{2} + 100\cdot 127^{3} + 109\cdot 127^{4} +O(127^{5})\)
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| $r_{ 8 }$ | $=$ |
\( 111 + 83\cdot 127 + 42\cdot 127^{2} + 82\cdot 127^{3} + 76\cdot 127^{4} +O(127^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,4)(5,7)(6,8)$ | $-2$ |
| $4$ | $2$ | $(1,8)(2,6)(3,5)(4,7)$ | $0$ |
| $4$ | $2$ | $(1,4)(2,3)(5,7)$ | $0$ |
| $2$ | $4$ | $(1,3,2,4)(5,8,7,6)$ | $0$ |
| $2$ | $8$ | $(1,8,4,5,2,6,3,7)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,5,3,8,2,7,4,6)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.