Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8$ |
| Conductor: | \(255025\)\(\medspace = 5^{2} \cdot 101^{2} \) |
| Frobenius-Schur indicator: | $-1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.8.16586252353140625.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $Q_8$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{5}, \sqrt{101})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} - 164x^{6} + 139x^{5} + 6881x^{4} - 11125x^{3} - 63850x^{2} + 149875x - 56125 \)
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The roots of $f$ are computed in $\Q_{ 71 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 1 + 29\cdot 71 + 47\cdot 71^{2} + 69\cdot 71^{3} + 27\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 10 + 51\cdot 71 + 38\cdot 71^{2} + 21\cdot 71^{3} + 67\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 17 + 43\cdot 71 + 19\cdot 71^{2} + 40\cdot 71^{3} + 47\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 25 + 70\cdot 71 + 14\cdot 71^{2} + 41\cdot 71^{3} + 22\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 28 + 2\cdot 71 + 11\cdot 71^{2} + 46\cdot 71^{3} + 48\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 40 + 61\cdot 71 + 23\cdot 71^{2} + 32\cdot 71^{3} + 43\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 41 + 8\cdot 71 + 54\cdot 71^{2} + 49\cdot 71^{3} + 39\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 52 + 17\cdot 71 + 3\cdot 71^{2} + 54\cdot 71^{3} + 57\cdot 71^{4} +O(71^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ | ✓ |
| $1$ | $2$ | $(1,8)(2,5)(3,4)(6,7)$ | $-2$ | |
| $2$ | $4$ | $(1,6,8,7)(2,4,5,3)$ | $0$ | |
| $2$ | $4$ | $(1,4,8,3)(2,7,5,6)$ | $0$ | |
| $2$ | $4$ | $(1,2,8,5)(3,7,4,6)$ | $0$ |