Basic invariants
| Dimension: | $2$ |
| Group: | $D_{8}$ |
| Conductor: | \(2952\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 41 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.205797003264.3 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $D_{8}$ |
| Parity: | odd |
| Determinant: | 1.328.2t1.b.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.23616.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} - 6x^{6} + 32x^{5} + 43x^{4} - 96x^{3} - 282x^{2} + 120x + 600 \)
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The roots of $f$ are computed in $\Q_{ 401 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 + 7\cdot 401 + 85\cdot 401^{2} + 341\cdot 401^{3} + 303\cdot 401^{4} + 241\cdot 401^{5} +O(401^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 23 + 376\cdot 401 + 71\cdot 401^{2} + 241\cdot 401^{3} + 152\cdot 401^{4} + 38\cdot 401^{5} +O(401^{6})\)
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| $r_{ 3 }$ | $=$ |
\( 109 + 94\cdot 401 + 237\cdot 401^{2} + 72\cdot 401^{3} + 96\cdot 401^{4} + 140\cdot 401^{5} +O(401^{6})\)
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| $r_{ 4 }$ | $=$ |
\( 121 + 144\cdot 401 + 153\cdot 401^{2} + 188\cdot 401^{3} + 214\cdot 401^{4} + 64\cdot 401^{5} +O(401^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 140 + 112\cdot 401 + 140\cdot 401^{2} + 88\cdot 401^{3} + 63\cdot 401^{4} + 262\cdot 401^{5} +O(401^{6})\)
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| $r_{ 6 }$ | $=$ |
\( 150 + 187\cdot 401 + 339\cdot 401^{2} + 299\cdot 401^{3} + 338\cdot 401^{4} + 157\cdot 401^{5} +O(401^{6})\)
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| $r_{ 7 }$ | $=$ |
\( 313 + 146\cdot 401 + 373\cdot 401^{2} + 32\cdot 401^{3} + 184\cdot 401^{4} + 218\cdot 401^{5} +O(401^{6})\)
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| $r_{ 8 }$ | $=$ |
\( 347 + 134\cdot 401 + 203\cdot 401^{2} + 339\cdot 401^{3} + 250\cdot 401^{4} + 79\cdot 401^{5} +O(401^{6})\)
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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 | Complex conjugation |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,5)(2,4)(3,6)(7,8)$ | $-2$ | |
| $4$ | $2$ | $(1,4)(2,5)(7,8)$ | $0$ | |
| $4$ | $2$ | $(1,8)(2,3)(4,6)(5,7)$ | $0$ | ✓ |
| $2$ | $4$ | $(1,4,5,2)(3,7,6,8)$ | $0$ | |
| $2$ | $8$ | $(1,6,4,8,5,3,2,7)$ | $-\zeta_{8}^{3} + \zeta_{8}$ | |
| $2$ | $8$ | $(1,8,2,6,5,7,4,3)$ | $\zeta_{8}^{3} - \zeta_{8}$ |