Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8$ |
| Conductor: | \(21025\)\(\medspace = 5^{2} \cdot 29^{2} \) |
| Frobenius-Schur indicator: | $-1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.8.9294114390625.1 |
| 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 \(\Q(\sqrt{5}, \sqrt{29})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} - 47x^{6} + 40x^{5} + 581x^{4} - 220x^{3} - 2038x^{2} - 932x - 109 \)
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The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 7\cdot 59 + 51\cdot 59^{2} + 13\cdot 59^{3} + 4\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 3 + 27\cdot 59 + 36\cdot 59^{2} + 42\cdot 59^{3} + 34\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 4 + 38\cdot 59 + 40\cdot 59^{2} + 43\cdot 59^{3} + 49\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 9 + 55\cdot 59 + 59^{2} + 31\cdot 59^{3} + 42\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 12 + 30\cdot 59 + 19\cdot 59^{2} + 48\cdot 59^{3} + 3\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 19 + 23\cdot 59 + 57\cdot 59^{2} + 42\cdot 59^{3} + 43\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 28 + 42\cdot 59 + 39\cdot 59^{2} + 35\cdot 59^{3} + 41\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 42 + 12\cdot 59 + 48\cdot 59^{2} + 36\cdot 59^{3} + 15\cdot 59^{4} +O(59^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ | ✓ |
| $1$ | $2$ | $(1,3)(2,5)(4,6)(7,8)$ | $-2$ | |
| $2$ | $4$ | $(1,7,3,8)(2,4,5,6)$ | $0$ | |
| $2$ | $4$ | $(1,4,3,6)(2,8,5,7)$ | $0$ | |
| $2$ | $4$ | $(1,2,3,5)(4,7,6,8)$ | $0$ |