Basic invariants
| Dimension: | $2$ |
| Group: | $\textrm{GL(2,3)}$ |
| Conductor: | \(2563\)\(\medspace = 11 \cdot 233 \) |
| Artin stem field: | Galois closure of 8.2.16836267547.4 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | 24T22 |
| Parity: | odd |
| Determinant: | 1.2563.2t1.a.a |
| Projective image: | $S_4$ |
| Projective stem field: | Galois closure of 4.2.2563.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} - 3x^{6} + x^{5} + 8x^{4} + 2x^{3} - 17x^{2} - 49x + 41 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$:
\( x^{2} + 63x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 55 a + 64 + \left(30 a + 8\right)\cdot 67 + \left(4 a + 57\right)\cdot 67^{2} + \left(13 a + 12\right)\cdot 67^{3} + \left(11 a + 22\right)\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 64 a + 54 + \left(51 a + 43\right)\cdot 67 + \left(11 a + 30\right)\cdot 67^{2} + \left(18 a + 53\right)\cdot 67^{3} + \left(38 a + 11\right)\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 25 + 49\cdot 67 + 43\cdot 67^{2} + 22\cdot 67^{3} + 3\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 3 a + 42 + \left(15 a + 53\right)\cdot 67 + \left(55 a + 25\right)\cdot 67^{2} + \left(48 a + 47\right)\cdot 67^{3} + \left(28 a + 12\right)\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 51 a + 25 + \left(58 a + 54\right)\cdot 67 + \left(63 a + 5\right)\cdot 67^{2} + \left(19 a + 42\right)\cdot 67^{3} + \left(28 a + 9\right)\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 15 + 10\cdot 67 + 59\cdot 67^{2} + 37\cdot 67^{3} + 51\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 12 a + 16 + \left(36 a + 10\right)\cdot 67 + \left(62 a + 44\right)\cdot 67^{2} + \left(53 a + 60\right)\cdot 67^{3} + \left(55 a + 53\right)\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 16 a + 28 + \left(8 a + 37\right)\cdot 67 + \left(3 a + 1\right)\cdot 67^{2} + \left(47 a + 58\right)\cdot 67^{3} + \left(38 a + 35\right)\cdot 67^{4} +O(67^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,5)(3,6)(4,8)$ | $-2$ |
| $12$ | $2$ | $(1,8)(3,6)(4,7)$ | $0$ |
| $8$ | $3$ | $(2,4,3)(5,8,6)$ | $-1$ |
| $6$ | $4$ | $(1,3,7,6)(2,4,5,8)$ | $0$ |
| $8$ | $6$ | $(1,7)(2,6,4,5,3,8)$ | $1$ |
| $6$ | $8$ | $(1,2,4,6,7,5,8,3)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ | $8$ | $(1,5,4,3,7,2,8,6)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.