Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(1400\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 7 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.19208000000.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.56.2t1.b.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.9800.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} + 9x^{6} - 27x^{5} + 44x^{4} - 94x^{3} + 156x^{2} - 88x + 16 \)
|
The roots of $f$ are computed in $\Q_{ 193 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 26 + 125\cdot 193 + 123\cdot 193^{2} + 186\cdot 193^{3} + 120\cdot 193^{4} + 38\cdot 193^{5} +O(193^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 37 + 146\cdot 193 + 183\cdot 193^{2} + 85\cdot 193^{3} + 36\cdot 193^{4} + 61\cdot 193^{5} +O(193^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 49 + 51\cdot 193 + 157\cdot 193^{2} + 15\cdot 193^{3} + 90\cdot 193^{4} + 6\cdot 193^{5} +O(193^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 51 + 61\cdot 193 + 67\cdot 193^{2} + 133\cdot 193^{3} + 5\cdot 193^{4} +O(193^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 59 + 82\cdot 193 + 180\cdot 193^{2} + 114\cdot 193^{3} + 92\cdot 193^{4} + 92\cdot 193^{5} +O(193^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 86 + 29\cdot 193 + 154\cdot 193^{2} + 175\cdot 193^{3} + 77\cdot 193^{4} + 77\cdot 193^{5} +O(193^{6})\)
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| $r_{ 7 }$ | $=$ |
\( 130 + 136\cdot 193 + 88\cdot 193^{2} + 83\cdot 193^{3} + 57\cdot 193^{4} + 36\cdot 193^{5} +O(193^{6})\)
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| $r_{ 8 }$ | $=$ |
\( 142 + 139\cdot 193 + 9\cdot 193^{2} + 169\cdot 193^{3} + 97\cdot 193^{4} + 73\cdot 193^{5} +O(193^{6})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,3)(4,7)(6,8)$ | $-2$ |
| $4$ | $2$ | $(1,7)(2,8)(3,6)(4,5)$ | $0$ |
| $4$ | $2$ | $(1,6)(4,7)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,6,5,8)(2,4,3,7)$ | $0$ |
| $2$ | $8$ | $(1,4,8,2,5,7,6,3)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,2,6,4,5,3,8,7)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.