Basic invariants
| Dimension: | $4$ |
| Group: | $\textrm{GL(2,3)}$ |
| Conductor: | \(4393216\)\(\medspace = 2^{8} \cdot 131^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.9208180736.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $\textrm{GL(2,3)}$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.2096.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} - 6x^{6} + 28x^{5} - 44x^{4} + 38x^{3} - 8x^{2} - 4x - 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( a + 28 + \left(18 a + 1\right)\cdot 29 + \left(26 a + 18\right)\cdot 29^{2} + \left(22 a + 15\right)\cdot 29^{3} + \left(4 a + 8\right)\cdot 29^{4} + \left(a + 13\right)\cdot 29^{5} + \left(7 a + 26\right)\cdot 29^{6} +O(29^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 2 a + 9 + \left(19 a + 10\right)\cdot 29 + \left(28 a + 12\right)\cdot 29^{2} + \left(20 a + 15\right)\cdot 29^{3} + \left(9 a + 21\right)\cdot 29^{4} + \left(6 a + 2\right)\cdot 29^{5} + \left(18 a + 19\right)\cdot 29^{6} +O(29^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 12 + 16\cdot 29 + 29^{2} + 13\cdot 29^{3} + 17\cdot 29^{4} + 22\cdot 29^{5} + 8\cdot 29^{6} +O(29^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 28 a + 4 + \left(10 a + 4\right)\cdot 29 + \left(2 a + 17\right)\cdot 29^{2} + \left(6 a + 16\right)\cdot 29^{3} + \left(24 a + 9\right)\cdot 29^{4} + \left(27 a + 14\right)\cdot 29^{5} + \left(21 a + 2\right)\cdot 29^{6} +O(29^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 18 a + 11 + \left(13 a + 14\right)\cdot 29 + \left(16 a + 15\right)\cdot 29^{2} + \left(4 a + 2\right)\cdot 29^{3} + \left(22 a + 8\right)\cdot 29^{4} + \left(26 a + 11\right)\cdot 29^{5} + \left(9 a + 4\right)\cdot 29^{6} +O(29^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 21 + 16\cdot 29 + 4\cdot 29^{2} + 10\cdot 29^{3} + 3\cdot 29^{4} + 20\cdot 29^{5} + 10\cdot 29^{6} +O(29^{7})\)
|
| $r_{ 7 }$ | $=$ |
\( 27 a + 19 + \left(9 a + 16\right)\cdot 29 + 20\cdot 29^{2} + \left(8 a + 4\right)\cdot 29^{3} + \left(19 a + 20\right)\cdot 29^{4} + \left(22 a + 24\right)\cdot 29^{5} + \left(10 a + 16\right)\cdot 29^{6} +O(29^{7})\)
|
| $r_{ 8 }$ | $=$ |
\( 11 a + 14 + \left(15 a + 6\right)\cdot 29 + \left(12 a + 26\right)\cdot 29^{2} + \left(24 a + 8\right)\cdot 29^{3} + \left(6 a + 27\right)\cdot 29^{4} + \left(2 a + 6\right)\cdot 29^{5} + \left(19 a + 27\right)\cdot 29^{6} +O(29^{7})\)
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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 |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,5)(2,7)(3,6)(4,8)$ | $-4$ |
| $12$ | $2$ | $(1,6)(3,5)(4,8)$ | $0$ |
| $8$ | $3$ | $(1,6,7)(2,5,3)$ | $1$ |
| $6$ | $4$ | $(1,2,5,7)(3,4,6,8)$ | $0$ |
| $8$ | $6$ | $(1,2,6,5,7,3)(4,8)$ | $-1$ |
| $6$ | $8$ | $(1,2,3,8,5,7,6,4)$ | $0$ |
| $6$ | $8$ | $(1,7,3,4,5,2,6,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.