Basic invariants
| Dimension: | $2$ |
| Group: | $D_4$ |
| Conductor: | \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.0.2985984.1 |
| 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 \(\Q(\zeta_{12})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + 2x^{6} - 2x^{5} + 7x^{4} - 10x^{3} + 8x^{2} - 4x + 1 \)
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The roots of $f$ are computed in $\Q_{ 37 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 10 + 32\cdot 37 + 27\cdot 37^{2} + 11\cdot 37^{3} + 22\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 14 + 35\cdot 37 + 27\cdot 37^{2} + 22\cdot 37^{3} + 34\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 19 + 2\cdot 37 + 9\cdot 37^{2} + 9\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 21 + 31\cdot 37 + 15\cdot 37^{2} + 34\cdot 37^{3} + 4\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 22 + 6\cdot 37 + 31\cdot 37^{2} + 2\cdot 37^{4} +O(37^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 31 + 10\cdot 37 + 14\cdot 37^{2} + 6\cdot 37^{3} + 4\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 34 + 37 + 34\cdot 37^{2} + 17\cdot 37^{3} + 22\cdot 37^{4} +O(37^{5})\)
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| $r_{ 8 }$ | $=$ |
\( 36 + 26\cdot 37 + 24\cdot 37^{2} + 16\cdot 37^{3} + 11\cdot 37^{4} +O(37^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,8)(3,4)(5,7)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,3)(4,8)(5,6)$ | $0$ |
| $2$ | $4$ | $(1,3,6,4)(2,7,8,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.