Basic invariants
| Dimension: | $4$ |
| Group: | $\textrm{GL(2,3)}$ |
| Conductor: | \(4765489\)\(\medspace = 37^{2} \cdot 59^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.10403062487.1 |
| 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 4.2.2183.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} - 8x^{6} + 15x^{5} + 20x^{4} - 46x^{3} - 17x^{2} + 74x - 37 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 79 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 79 }$:
\( x^{2} + 78x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 + 33\cdot 79 + 23\cdot 79^{2} + 61\cdot 79^{3} + 9\cdot 79^{4} + 27\cdot 79^{5} +O(79^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 6 a + 77 + \left(36 a + 1\right)\cdot 79 + \left(23 a + 17\right)\cdot 79^{2} + \left(64 a + 39\right)\cdot 79^{3} + \left(75 a + 75\right)\cdot 79^{4} + \left(25 a + 56\right)\cdot 79^{5} +O(79^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 35 a + 17 + \left(18 a + 17\right)\cdot 79 + \left(6 a + 41\right)\cdot 79^{2} + \left(17 a + 51\right)\cdot 79^{3} + \left(42 a + 74\right)\cdot 79^{4} + \left(42 a + 59\right)\cdot 79^{5} +O(79^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 51 a + 27 + \left(68 a + 32\right)\cdot 79 + \left(33 a + 59\right)\cdot 79^{2} + \left(54 a + 64\right)\cdot 79^{3} + \left(66 a + 51\right)\cdot 79^{4} + \left(75 a + 59\right)\cdot 79^{5} +O(79^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 73 a + 4 + \left(42 a + 32\right)\cdot 79 + \left(55 a + 4\right)\cdot 79^{2} + \left(14 a + 1\right)\cdot 79^{3} + \left(3 a + 8\right)\cdot 79^{4} + \left(53 a + 7\right)\cdot 79^{5} +O(79^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 44 a + 52 + 60 a\cdot 79 + \left(72 a + 29\right)\cdot 79^{2} + \left(61 a + 62\right)\cdot 79^{3} + \left(36 a + 20\right)\cdot 79^{4} + \left(36 a + 60\right)\cdot 79^{5} +O(79^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 56 + 69\cdot 79 + 37\cdot 79^{2} + 29\cdot 79^{3} + 11\cdot 79^{4} + 55\cdot 79^{5} +O(79^{6})\)
|
| $r_{ 8 }$ | $=$ |
\( 28 a + 78 + \left(10 a + 49\right)\cdot 79 + \left(45 a + 24\right)\cdot 79^{2} + \left(24 a + 6\right)\cdot 79^{3} + \left(12 a + 64\right)\cdot 79^{4} + \left(3 a + 68\right)\cdot 79^{5} +O(79^{6})\)
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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 |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,7)(2,3)(4,8)(5,6)$ | $-4$ |
| $12$ | $2$ | $(2,3)(4,5)(6,8)$ | $0$ |
| $8$ | $3$ | $(1,6,8)(4,7,5)$ | $1$ |
| $6$ | $4$ | $(1,4,7,8)(2,6,3,5)$ | $0$ |
| $8$ | $6$ | $(1,7)(2,8,6,3,4,5)$ | $-1$ |
| $6$ | $8$ | $(1,5,3,4,7,6,2,8)$ | $0$ |
| $6$ | $8$ | $(1,6,3,8,7,5,2,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.