Basic invariants
| Dimension: | $4$ |
| Group: | $\textrm{GL(2,3)}$ |
| Conductor: | \(3323329\)\(\medspace = 1823^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.6058428767.1 |
| 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 4.2.1823.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + 7x^{6} - 6x^{5} - 4x^{4} - 3x^{3} + 6x^{2} - 5x - 19 \)
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The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 26 + 7\cdot 31 + 27\cdot 31^{2} + 24\cdot 31^{4} + 22\cdot 31^{5} + 9\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 10 a + 6 + \left(8 a + 29\right)\cdot 31 + 30\cdot 31^{2} + \left(19 a + 8\right)\cdot 31^{3} + \left(27 a + 18\right)\cdot 31^{4} + \left(11 a + 1\right)\cdot 31^{5} + \left(28 a + 27\right)\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 21 a + 26 + \left(22 a + 4\right)\cdot 31 + \left(30 a + 23\right)\cdot 31^{2} + \left(11 a + 15\right)\cdot 31^{3} + \left(3 a + 23\right)\cdot 31^{4} + \left(19 a + 28\right)\cdot 31^{5} + \left(2 a + 9\right)\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 2 a + 25 + \left(23 a + 1\right)\cdot 31 + \left(24 a + 4\right)\cdot 31^{2} + \left(2 a + 8\right)\cdot 31^{3} + \left(4 a + 10\right)\cdot 31^{4} + \left(10 a + 4\right)\cdot 31^{5} + \left(a + 8\right)\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 3 a + 4 + \left(21 a + 26\right)\cdot 31 + \left(9 a + 30\right)\cdot 31^{2} + \left(7 a + 12\right)\cdot 31^{3} + \left(30 a + 7\right)\cdot 31^{4} + \left(7 a + 16\right)\cdot 31^{5} + \left(6 a + 16\right)\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 2 + 5\cdot 31 + 10\cdot 31^{2} + 8\cdot 31^{3} + 26\cdot 31^{4} + 27\cdot 31^{5} + 30\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 7 }$ | $=$ |
\( 28 a + 10 + \left(9 a + 3\right)\cdot 31 + \left(21 a + 29\right)\cdot 31^{2} + \left(23 a + 17\right)\cdot 31^{3} + 29\cdot 31^{4} + \left(23 a + 1\right)\cdot 31^{5} + \left(24 a + 21\right)\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 8 }$ | $=$ |
\( 29 a + 29 + \left(7 a + 14\right)\cdot 31 + \left(6 a + 30\right)\cdot 31^{2} + \left(28 a + 19\right)\cdot 31^{3} + \left(26 a + 15\right)\cdot 31^{4} + \left(20 a + 20\right)\cdot 31^{5} + 29 a\cdot 31^{6} +O(31^{7})\)
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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 |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,6)(2,4)(3,8)(5,7)$ | $-4$ |
| $12$ | $2$ | $(1,7)(3,8)(5,6)$ | $0$ |
| $8$ | $3$ | $(1,8,4)(2,6,3)$ | $1$ |
| $6$ | $4$ | $(1,7,6,5)(2,8,4,3)$ | $0$ |
| $8$ | $6$ | $(1,2,8,6,4,3)(5,7)$ | $-1$ |
| $6$ | $8$ | $(1,4,5,8,6,2,7,3)$ | $0$ |
| $6$ | $8$ | $(1,2,5,3,6,4,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.