Basic invariants
| Dimension: | $2$ |
| Group: | $D_4$ |
| Conductor: | \(221\)\(\medspace = 13 \cdot 17 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.0.2385443281.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | even |
| Determinant: | 1.221.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{13}, \sqrt{17})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + x^{6} - 4x^{5} - 38x^{4} - 2x^{3} + 123x^{2} - 34x + 17 \)
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The roots of $f$ are computed in $\Q_{ 103 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 8 + 98\cdot 103 + 44\cdot 103^{2} + 47\cdot 103^{3} + 36\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 26 + 45\cdot 103 + 43\cdot 103^{2} + 23\cdot 103^{3} + 47\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 30 + 87\cdot 103 + 21\cdot 103^{2} + 20\cdot 103^{3} + 20\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 50 + 31\cdot 103 + 91\cdot 103^{2} + 95\cdot 103^{3} + 13\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 57 + 17\cdot 103 + 20\cdot 103^{2} + 40\cdot 103^{3} + 99\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 64 + 56\cdot 103 + 30\cdot 103^{2} + 39\cdot 103^{3} + 76\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 84 + 19\cdot 103 + 39\cdot 103^{2} + 23\cdot 103^{3} + 79\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 93 + 55\cdot 103 + 17\cdot 103^{2} + 19\cdot 103^{3} + 39\cdot 103^{4} +O(103^{5})\)
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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 | Complex conjugation |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,4)(2,8)(3,5)(6,7)$ | $-2$ | ✓ |
| $2$ | $2$ | $(1,2)(3,6)(4,8)(5,7)$ | $0$ | |
| $2$ | $2$ | $(1,3)(2,7)(4,5)(6,8)$ | $0$ | |
| $2$ | $4$ | $(1,7,4,6)(2,3,8,5)$ | $0$ |