Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(399\)\(\medspace = 3 \cdot 7 \cdot 19 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.190563597.4 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.399.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.1197.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} + 7x^{6} - 6x^{5} + x^{4} + 6x^{3} - 3x^{2} - 9x + 9 \)
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The roots of $f$ are computed in $\Q_{ 181 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 22 + 175\cdot 181 + 79\cdot 181^{2} + 7\cdot 181^{3} + 73\cdot 181^{4} +O(181^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 75 + 83\cdot 181 + 155\cdot 181^{2} + 167\cdot 181^{3} + 175\cdot 181^{4} +O(181^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 78 + 20\cdot 181 + 15\cdot 181^{2} + 167\cdot 181^{3} + 131\cdot 181^{4} +O(181^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 79 + 152\cdot 181 + 68\cdot 181^{2} + 50\cdot 181^{3} + 24\cdot 181^{4} +O(181^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 93 + 92\cdot 181 + 125\cdot 181^{2} + 34\cdot 181^{3} + 158\cdot 181^{4} +O(181^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 95 + 110\cdot 181 + 92\cdot 181^{2} + 35\cdot 181^{3} + 103\cdot 181^{4} +O(181^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 119 + 106\cdot 181 + 23\cdot 181^{2} + 91\cdot 181^{3} + 51\cdot 181^{4} +O(181^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 166 + 163\cdot 181 + 162\cdot 181^{2} + 169\cdot 181^{3} + 5\cdot 181^{4} +O(181^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,5)(3,8)(4,6)$ | $-2$ |
| $4$ | $2$ | $(1,4)(2,5)(6,7)$ | $0$ |
| $4$ | $2$ | $(1,2)(3,6)(4,8)(5,7)$ | $0$ |
| $2$ | $4$ | $(1,4,7,6)(2,3,5,8)$ | $0$ |
| $2$ | $8$ | $(1,8,4,2,7,3,6,5)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,2,6,8,7,5,4,3)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.