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 8.2.10403062487.2 |
| 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
| $f(x)$ | $=$ |
\( x^{8} - x^{7} - 2x^{6} - 9x^{5} - 31x^{4} - 52x^{3} - 47x^{2} - 58x - 28 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 4 a + 43 + \left(63 a + 71\right)\cdot 79 + \left(a + 44\right)\cdot 79^{2} + \left(37 a + 76\right)\cdot 79^{3} + \left(12 a + 49\right)\cdot 79^{4} + \left(2 a + 11\right)\cdot 79^{5} +O(79^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 24 + 71\cdot 79 + 67\cdot 79^{2} + 63\cdot 79^{3} + 25\cdot 79^{4} + 44\cdot 79^{5} +O(79^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 13 a + 51 + \left(7 a + 30\right)\cdot 79 + \left(28 a + 13\right)\cdot 79^{2} + \left(a + 7\right)\cdot 79^{3} + \left(62 a + 40\right)\cdot 79^{4} + \left(2 a + 7\right)\cdot 79^{5} +O(79^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 40 a + 56 + \left(48 a + 36\right)\cdot 79 + \left(64 a + 30\right)\cdot 79^{2} + \left(66 a + 43\right)\cdot 79^{3} + \left(37 a + 55\right)\cdot 79^{4} + \left(59 a + 42\right)\cdot 79^{5} +O(79^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 75 a + 47 + \left(15 a + 51\right)\cdot 79 + \left(77 a + 62\right)\cdot 79^{2} + \left(41 a + 32\right)\cdot 79^{3} + \left(66 a + 25\right)\cdot 79^{4} + \left(76 a + 1\right)\cdot 79^{5} +O(79^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 15 + 63\cdot 79 + 15\cdot 79^{2} + 66\cdot 79^{3} + 70\cdot 79^{4} + 37\cdot 79^{5} +O(79^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 66 a + 64 + \left(71 a + 24\right)\cdot 79 + \left(50 a + 34\right)\cdot 79^{2} + \left(77 a + 59\right)\cdot 79^{3} + \left(16 a + 21\right)\cdot 79^{4} + \left(76 a + 27\right)\cdot 79^{5} +O(79^{6})\)
|
| $r_{ 8 }$ | $=$ |
\( 39 a + 17 + \left(30 a + 45\right)\cdot 79 + \left(14 a + 46\right)\cdot 79^{2} + \left(12 a + 45\right)\cdot 79^{3} + \left(41 a + 26\right)\cdot 79^{4} + \left(19 a + 64\right)\cdot 79^{5} +O(79^{6})\)
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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 |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,6)(3,7)(4,5)$ | $-4$ |
| $12$ | $2$ | $(1,8)(2,7)(3,6)$ | $0$ |
| $8$ | $3$ | $(1,5,6)(2,8,4)$ | $1$ |
| $6$ | $4$ | $(1,3,8,7)(2,5,6,4)$ | $0$ |
| $8$ | $6$ | $(1,2,5,8,6,4)(3,7)$ | $-1$ |
| $6$ | $8$ | $(1,7,5,6,8,3,4,2)$ | $0$ |
| $6$ | $8$ | $(1,3,5,2,8,7,4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.