Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(299\)\(\medspace = 13 \cdot 23 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.614810677.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.299.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.2.3887.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} + x^{6} - x^{4} - x^{3} + 7x^{2} + 2x + 8 \)
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The roots of $f$ are computed in $\Q_{ 131 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 12 + 2\cdot 131 + 114\cdot 131^{2} + 123\cdot 131^{3} + 120\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 14 + 32\cdot 131 + 49\cdot 131^{2} + 67\cdot 131^{3} + 26\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 24 + 82\cdot 131 + 71\cdot 131^{2} + 8\cdot 131^{3} + 89\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 58 + 127\cdot 131 + 31\cdot 131^{2} + 95\cdot 131^{3} + 4\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 85 + 27\cdot 131 + 63\cdot 131^{2} + 60\cdot 131^{3} + 101\cdot 131^{4} +O(131^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 96 + 62\cdot 131 + 46\cdot 131^{2} + 35\cdot 131^{3} + 128\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 109 + 120\cdot 131 + 37\cdot 131^{2} + 2\cdot 131^{3} + 31\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 127 + 68\cdot 131 + 109\cdot 131^{2} + 130\cdot 131^{3} + 21\cdot 131^{4} +O(131^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,4)(3,8)(5,7)$ | $-2$ |
| $4$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $0$ |
| $4$ | $2$ | $(1,6)(3,7)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,2,6,4)(3,7,8,5)$ | $0$ |
| $2$ | $8$ | $(1,5,4,8,6,7,2,3)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,8,2,5,6,3,4,7)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.