Basic invariants
| Dimension: | $4$ |
| Group: | $Q_8:S_4$ |
| Conductor: | \(518400\)\(\medspace = 2^{8} \cdot 3^{4} \cdot 5^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.3359232000.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $Q_8:S_4$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $C_2^2:S_4$ |
| Projective stem field: | Galois closure of 8.4.26244000000.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 4x^{6} + 12x^{4} + 20x^{2} + 20 \)
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The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$:
\( x^{2} + 12x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 7 a + 3 + \left(11 a + 4\right)\cdot 13 + 10 a\cdot 13^{2} + \left(9 a + 7\right)\cdot 13^{3} + \left(4 a + 2\right)\cdot 13^{4} + \left(9 a + 4\right)\cdot 13^{5} + \left(3 a + 9\right)\cdot 13^{6} + 13^{7} + \left(3 a + 5\right)\cdot 13^{8} + \left(3 a + 6\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 4 + 2\cdot 13 + 13^{2} + 13^{3} + 8\cdot 13^{4} + 7\cdot 13^{6} + 12\cdot 13^{7} + 9\cdot 13^{8} + 4\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 5 a + 10 + \left(8 a + 4\right)\cdot 13 + \left(9 a + 8\right)\cdot 13^{2} + \left(4 a + 4\right)\cdot 13^{3} + \left(2 a + 8\right)\cdot 13^{4} + \left(6 a + 6\right)\cdot 13^{5} + \left(9 a + 7\right)\cdot 13^{6} + \left(9 a + 12\right)\cdot 13^{7} + 6\cdot 13^{8} + \left(12 a + 9\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 8 a + 2 + \left(4 a + 8\right)\cdot 13 + \left(3 a + 9\right)\cdot 13^{2} + \left(8 a + 12\right)\cdot 13^{3} + \left(10 a + 5\right)\cdot 13^{4} + \left(6 a + 10\right)\cdot 13^{5} + \left(3 a + 10\right)\cdot 13^{6} + \left(3 a + 12\right)\cdot 13^{7} + \left(12 a + 10\right)\cdot 13^{8} + 7\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 6 a + 10 + \left(a + 8\right)\cdot 13 + \left(2 a + 12\right)\cdot 13^{2} + \left(3 a + 5\right)\cdot 13^{3} + \left(8 a + 10\right)\cdot 13^{4} + \left(3 a + 8\right)\cdot 13^{5} + \left(9 a + 3\right)\cdot 13^{6} + \left(12 a + 11\right)\cdot 13^{7} + \left(9 a + 7\right)\cdot 13^{8} + \left(9 a + 6\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 9 + 10\cdot 13 + 11\cdot 13^{2} + 11\cdot 13^{3} + 4\cdot 13^{4} + 12\cdot 13^{5} + 5\cdot 13^{6} + 3\cdot 13^{8} + 8\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 8 a + 3 + \left(4 a + 8\right)\cdot 13 + \left(3 a + 4\right)\cdot 13^{2} + \left(8 a + 8\right)\cdot 13^{3} + \left(10 a + 4\right)\cdot 13^{4} + \left(6 a + 6\right)\cdot 13^{5} + \left(3 a + 5\right)\cdot 13^{6} + 3 a\cdot 13^{7} + \left(12 a + 6\right)\cdot 13^{8} + 3\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 5 a + 11 + \left(8 a + 4\right)\cdot 13 + \left(9 a + 3\right)\cdot 13^{2} + 4 a\cdot 13^{3} + \left(2 a + 7\right)\cdot 13^{4} + \left(6 a + 2\right)\cdot 13^{5} + \left(9 a + 2\right)\cdot 13^{6} + 9 a\cdot 13^{7} + 2\cdot 13^{8} + \left(12 a + 5\right)\cdot 13^{9} +O(13^{10})\)
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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 |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $-4$ |
| $6$ | $2$ | $(1,5)(2,6)$ | $0$ |
| $12$ | $2$ | $(1,3)(2,8)(4,6)(5,7)$ | $0$ |
| $24$ | $2$ | $(1,2)(3,7)(5,6)$ | $0$ |
| $32$ | $3$ | $(2,3,8)(4,6,7)$ | $1$ |
| $6$ | $4$ | $(1,3,5,7)(2,8,6,4)$ | $0$ |
| $6$ | $4$ | $(1,3,5,7)(2,4,6,8)$ | $0$ |
| $12$ | $4$ | $(1,6,5,2)(3,7)(4,8)$ | $-2$ |
| $12$ | $4$ | $(2,8,6,4)$ | $2$ |
| $32$ | $6$ | $(1,5)(2,4,3,6,8,7)$ | $-1$ |
| $24$ | $8$ | $(1,8,3,6,5,4,7,2)$ | $0$ |
| $24$ | $8$ | $(1,7,4,6,5,3,8,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.