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 8.8.1340095640625.1 |
| 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
| $f(x)$ | $=$ |
\( x^{8} - x^{7} - 34x^{6} + 29x^{5} + 361x^{4} - 305x^{3} - 1090x^{2} + 1345x - 395 \)
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The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 42\cdot 79 + 27\cdot 79^{2} + 30\cdot 79^{3} + 73\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 7 + 63\cdot 79 + 39\cdot 79^{2} + 76\cdot 79^{3} + 39\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 25 + 29\cdot 79 + 38\cdot 79^{2} + 5\cdot 79^{3} + 26\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 45 + 41\cdot 79 + 21\cdot 79^{2} + 49\cdot 79^{3} + 25\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 49 + 16\cdot 79 + 55\cdot 79^{2} + 15\cdot 79^{3} + 37\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 53 + 79 + 12\cdot 79^{2} + 68\cdot 79^{3} + 15\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 66 + 72\cdot 79 + 42\cdot 79^{2} + 11\cdot 79^{3} + 57\cdot 79^{4} +O(79^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 72 + 48\cdot 79 + 78\cdot 79^{2} + 58\cdot 79^{3} + 40\cdot 79^{4} +O(79^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ | ✓ |
| $1$ | $2$ | $(1,2)(3,7)(4,8)(5,6)$ | $-2$ | |
| $2$ | $4$ | $(1,8,2,4)(3,5,7,6)$ | $0$ | |
| $2$ | $4$ | $(1,5,2,6)(3,4,7,8)$ | $0$ | |
| $2$ | $4$ | $(1,3,2,7)(4,5,8,6)$ | $0$ |