Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8$ |
| Conductor: | \(42025\)\(\medspace = 5^{2} \cdot 41^{2} \) |
| Frobenius-Schur indicator: | $-1$ |
| Root number: | $-1$ |
| Artin field: | Galois closure of 8.8.74220378765625.1 |
| 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 \(\Q(\sqrt{5}, \sqrt{41})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} - 63x^{6} + 90x^{5} + 1311x^{4} - 20x^{3} - 7702x^{2} - 5524x - 1009 \)
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The roots of $f$ are computed in $\Q_{ 251 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 31 + 114\cdot 251^{2} + 222\cdot 251^{3} + 73\cdot 251^{4} +O(251^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 56 + 6\cdot 251 + 134\cdot 251^{2} + 68\cdot 251^{3} + 186\cdot 251^{4} +O(251^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 58 + 181\cdot 251 + 246\cdot 251^{2} + 82\cdot 251^{3} + 47\cdot 251^{4} +O(251^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 82 + 29\cdot 251 + 179\cdot 251^{2} + 198\cdot 251^{3} + 231\cdot 251^{4} +O(251^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 135 + 142\cdot 251 + 64\cdot 251^{2} + 167\cdot 251^{3} + 156\cdot 251^{4} +O(251^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 174 + 30\cdot 251 + 117\cdot 251^{2} + 189\cdot 251^{3} + 73\cdot 251^{4} +O(251^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 225 + 239\cdot 251 + 212\cdot 251^{2} + 77\cdot 251^{3} + 239\cdot 251^{4} +O(251^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 246 + 122\cdot 251 + 186\cdot 251^{2} + 247\cdot 251^{3} + 245\cdot 251^{4} +O(251^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ | ✓ |
| $1$ | $2$ | $(1,2)(3,7)(4,8)(5,6)$ | $-2$ | |
| $2$ | $4$ | $(1,8,2,4)(3,6,7,5)$ | $0$ | |
| $2$ | $4$ | $(1,7,2,3)(4,5,8,6)$ | $0$ | |
| $2$ | $4$ | $(1,6,2,5)(3,4,7,8)$ | $0$ |