Basic invariants
| Dimension: | $2$ |
| Group: | $\textrm{GL(2,3)}$ |
| Conductor: | \(1304\)\(\medspace = 2^{3} \cdot 163 \) |
| Artin number field: | Galois closure of 8.2.4434684928.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | 24T22 |
| Parity: | odd |
| Projective image: | $S_4$ |
| Projective field: | Galois closure of 4.2.2608.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$:
\( x^{2} + 63x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 26 a + 10 + \left(59 a + 40\right)\cdot 67 + 15\cdot 67^{2} + \left(16 a + 24\right)\cdot 67^{3} + \left(14 a + 12\right)\cdot 67^{4} + 17\cdot 67^{5} +O(67^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 9 + 18\cdot 67 + 50\cdot 67^{2} + 58\cdot 67^{3} + 28\cdot 67^{4} + 38\cdot 67^{5} +O(67^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 41 a + 47 + \left(7 a + 50\right)\cdot 67 + \left(66 a + 26\right)\cdot 67^{2} + \left(50 a + 20\right)\cdot 67^{3} + \left(52 a + 53\right)\cdot 67^{4} + \left(66 a + 3\right)\cdot 67^{5} +O(67^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 44 a + 19 + \left(60 a + 47\right)\cdot 67 + \left(41 a + 2\right)\cdot 67^{2} + \left(20 a + 66\right)\cdot 67^{3} + 46\cdot 67^{4} + \left(20 a + 9\right)\cdot 67^{5} +O(67^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 23 a + 61 + \left(6 a + 44\right)\cdot 67 + \left(25 a + 42\right)\cdot 67^{2} + \left(46 a + 39\right)\cdot 67^{3} + \left(66 a + 27\right)\cdot 67^{4} + \left(46 a + 22\right)\cdot 67^{5} +O(67^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 7 a + 5 + \left(33 a + 36\right)\cdot 67 + \left(18 a + 43\right)\cdot 67^{2} + \left(11 a + 2\right)\cdot 67^{3} + \left(33 a + 59\right)\cdot 67^{4} + \left(61 a + 47\right)\cdot 67^{5} +O(67^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 60 a + 33 + \left(33 a + 27\right)\cdot 67 + \left(48 a + 17\right)\cdot 67^{2} + \left(55 a + 29\right)\cdot 67^{3} + \left(33 a + 46\right)\cdot 67^{4} + \left(5 a + 59\right)\cdot 67^{5} +O(67^{6})\)
|
| $r_{ 8 }$ | $=$ |
\( 21 + 3\cdot 67 + 2\cdot 67^{2} + 27\cdot 67^{3} + 60\cdot 67^{4} + 67^{5} +O(67^{6})\)
|
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 values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,3)(2,8)(4,6)(5,7)$ | $-2$ | $-2$ |
| $12$ | $2$ | $(2,5)(4,6)(7,8)$ | $0$ | $0$ |
| $8$ | $3$ | $(1,7,4)(3,5,6)$ | $-1$ | $-1$ |
| $6$ | $4$ | $(1,6,3,4)(2,7,8,5)$ | $0$ | $0$ |
| $8$ | $6$ | $(1,8,5,3,2,7)(4,6)$ | $1$ | $1$ |
| $6$ | $8$ | $(1,8,4,7,3,2,6,5)$ | $-\zeta_{8}^{3} - \zeta_{8}$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ | $8$ | $(1,2,4,5,3,8,6,7)$ | $\zeta_{8}^{3} + \zeta_{8}$ | $-\zeta_{8}^{3} - \zeta_{8}$ |