Basic invariants
| Dimension: | $4$ |
| Group: | $\textrm{GL(2,3)}$ |
| Conductor: | \(1498176\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 17^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.11002604544.1 |
| 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 4.2.7344.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 6x^{4} - 24x^{2} - 51 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$:
\( x^{2} + 58x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 40 + 49\cdot 59 + 13\cdot 59^{2} + 5\cdot 59^{3} + 28\cdot 59^{4} + 54\cdot 59^{5} + 16\cdot 59^{6} + 26\cdot 59^{7} +O(59^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 8 a + 55 + \left(52 a + 36\right)\cdot 59 + \left(34 a + 8\right)\cdot 59^{2} + \left(4 a + 15\right)\cdot 59^{3} + \left(46 a + 38\right)\cdot 59^{4} + \left(11 a + 46\right)\cdot 59^{5} + \left(30 a + 49\right)\cdot 59^{6} + \left(45 a + 21\right)\cdot 59^{7} +O(59^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 8 a + 24 + \left(8 a + 33\right)\cdot 59 + \left(28 a + 17\right)\cdot 59^{2} + \left(40 a + 44\right)\cdot 59^{3} + \left(38 a + 26\right)\cdot 59^{4} + \left(29 a + 48\right)\cdot 59^{5} + \left(a + 53\right)\cdot 59^{6} + \left(7 a + 42\right)\cdot 59^{7} +O(59^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 8 a + 27 + \left(8 a + 25\right)\cdot 59 + \left(28 a + 21\right)\cdot 59^{2} + \left(40 a + 2\right)\cdot 59^{3} + \left(38 a + 34\right)\cdot 59^{4} + \left(29 a + 19\right)\cdot 59^{5} + \left(a + 33\right)\cdot 59^{6} + \left(7 a + 10\right)\cdot 59^{7} +O(59^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 19 + 9\cdot 59 + 45\cdot 59^{2} + 53\cdot 59^{3} + 30\cdot 59^{4} + 4\cdot 59^{5} + 42\cdot 59^{6} + 32\cdot 59^{7} +O(59^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( 51 a + 4 + \left(6 a + 22\right)\cdot 59 + \left(24 a + 50\right)\cdot 59^{2} + \left(54 a + 43\right)\cdot 59^{3} + \left(12 a + 20\right)\cdot 59^{4} + \left(47 a + 12\right)\cdot 59^{5} + \left(28 a + 9\right)\cdot 59^{6} + \left(13 a + 37\right)\cdot 59^{7} +O(59^{8})\)
|
| $r_{ 7 }$ | $=$ |
\( 51 a + 35 + \left(50 a + 25\right)\cdot 59 + \left(30 a + 41\right)\cdot 59^{2} + \left(18 a + 14\right)\cdot 59^{3} + \left(20 a + 32\right)\cdot 59^{4} + \left(29 a + 10\right)\cdot 59^{5} + \left(57 a + 5\right)\cdot 59^{6} + \left(51 a + 16\right)\cdot 59^{7} +O(59^{8})\)
|
| $r_{ 8 }$ | $=$ |
\( 51 a + 32 + \left(50 a + 33\right)\cdot 59 + \left(30 a + 37\right)\cdot 59^{2} + \left(18 a + 56\right)\cdot 59^{3} + \left(20 a + 24\right)\cdot 59^{4} + \left(29 a + 39\right)\cdot 59^{5} + \left(57 a + 25\right)\cdot 59^{6} + \left(51 a + 48\right)\cdot 59^{7} +O(59^{8})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $-4$ |
| $12$ | $2$ | $(1,5)(2,8)(4,6)$ | $0$ |
| $8$ | $3$ | $(1,6,8)(2,4,5)$ | $1$ |
| $6$ | $4$ | $(1,7,5,3)(2,4,6,8)$ | $0$ |
| $8$ | $6$ | $(1,8,7,5,4,3)(2,6)$ | $-1$ |
| $6$ | $8$ | $(1,4,3,2,5,8,7,6)$ | $0$ |
| $6$ | $8$ | $(1,8,3,6,5,4,7,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.