Basic invariants
| Dimension: | $2$ |
| Group: | $D_4$ |
| Conductor: | \(624\)\(\medspace = 2^{4} \cdot 3 \cdot 13 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.0.592240896.5 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.39.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{13})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + x^{6} + 4x^{4} - 3x^{2} + 9 \)
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The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 13 + 36\cdot 79 + 66\cdot 79^{2} + 18\cdot 79^{3} + 4\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 17 + 7\cdot 79 + 36\cdot 79^{2} + 65\cdot 79^{3} + 51\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 26 + 5\cdot 79 + 44\cdot 79^{2} + 68\cdot 79^{3} + 18\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 34 + 54\cdot 79 + 9\cdot 79^{2} + 5\cdot 79^{3} + 5\cdot 79^{4} +O(79^{5})\)
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| $r_{ 5 }$ | $=$ |
\( 45 + 24\cdot 79 + 69\cdot 79^{2} + 73\cdot 79^{3} + 73\cdot 79^{4} +O(79^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 53 + 73\cdot 79 + 34\cdot 79^{2} + 10\cdot 79^{3} + 60\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 62 + 71\cdot 79 + 42\cdot 79^{2} + 13\cdot 79^{3} + 27\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 66 + 42\cdot 79 + 12\cdot 79^{2} + 60\cdot 79^{3} + 74\cdot 79^{4} +O(79^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.