Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8$ |
| Conductor: | \(11025\)\(\medspace = 3^{2} \cdot 5^{2} \cdot 7^{2} \) |
| Frobenius-Schur indicator: | $-1$ |
| Root number: | $-1$ |
| Artin field: | Galois closure of 8.0.1340095640625.1 |
| 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 \(\Q(\sqrt{5}, \sqrt{21})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} + 22x^{6} - 60x^{5} + 201x^{4} - 450x^{3} + 1528x^{2} - 3069x + 4561 \)
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The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 13 + 43\cdot 79 + 24\cdot 79^{2} + 4\cdot 79^{3} + 13\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 14 + 4\cdot 79 + 56\cdot 79^{2} + 36\cdot 79^{3} + 63\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 24 + 15\cdot 79 + 18\cdot 79^{2} + 67\cdot 79^{4} +O(79^{5})\)
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| $r_{ 4 }$ | $=$ |
\( 29 + 24\cdot 79 + 33\cdot 79^{2} + 45\cdot 79^{3} + 78\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 49 + 37\cdot 79 + 72\cdot 79^{2} + 50\cdot 79^{3} + 56\cdot 79^{4} +O(79^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 54 + 51\cdot 79 + 20\cdot 79^{2} + 25\cdot 79^{3} + 11\cdot 79^{4} +O(79^{5})\)
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| $r_{ 7 }$ | $=$ |
\( 63 + 30\cdot 79 + 79^{2} + 12\cdot 79^{3} + 15\cdot 79^{4} +O(79^{5})\)
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| $r_{ 8 }$ | $=$ |
\( 73 + 29\cdot 79 + 10\cdot 79^{2} + 62\cdot 79^{3} + 10\cdot 79^{4} +O(79^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,4)(2,6)(3,5)(7,8)$ | $-2$ | ✓ |
| $2$ | $4$ | $(1,8,4,7)(2,3,6,5)$ | $0$ | |
| $2$ | $4$ | $(1,2,4,6)(3,8,5,7)$ | $0$ | |
| $2$ | $4$ | $(1,5,4,3)(2,8,6,7)$ | $0$ |