Basic invariants
| Dimension: | $2$ |
| Group: | $D_4$ |
| Conductor: | \(256\)\(\medspace = 2^{8} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.0.16777216.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.4.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\zeta_{8})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{6} + 8x^{4} - 4x^{2} + 1 \)
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The roots of $f$ are computed in $\Q_{ 73 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 5 + 72\cdot 73 + 34\cdot 73^{2} + 34\cdot 73^{3} + 49\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 19 + 66\cdot 73 + 37\cdot 73^{2} + 34\cdot 73^{3} + 18\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 23 + 12\cdot 73 + 16\cdot 73^{2} + 36\cdot 73^{3} + 2\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 29 + 20\cdot 73 + 49\cdot 73^{2} + 5\cdot 73^{3} + 7\cdot 73^{4} +O(73^{5})\)
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| $r_{ 5 }$ | $=$ |
\( 44 + 52\cdot 73 + 23\cdot 73^{2} + 67\cdot 73^{3} + 65\cdot 73^{4} +O(73^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 50 + 60\cdot 73 + 56\cdot 73^{2} + 36\cdot 73^{3} + 70\cdot 73^{4} +O(73^{5})\)
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| $r_{ 7 }$ | $=$ |
\( 54 + 6\cdot 73 + 35\cdot 73^{2} + 38\cdot 73^{3} + 54\cdot 73^{4} +O(73^{5})\)
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| $r_{ 8 }$ | $=$ |
\( 68 + 38\cdot 73^{2} + 38\cdot 73^{3} + 23\cdot 73^{4} +O(73^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.