Basic invariants
Dimension: | $2$ |
Group: | $\textrm{GL(2,3)}$ |
Conductor: | \(3600\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5^{2} \) |
Artin stem field: | Galois closure of 8.2.139968000000.2 |
Galois orbit size: | $2$ |
Smallest permutation container: | 24T22 |
Parity: | odd |
Determinant: | 1.3.2t1.a.a |
Projective image: | $S_4$ |
Projective stem field: | Galois closure of 4.2.10800.2 |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} + 60x^{2} - 300 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: \( x^{2} + 49x + 2 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 3 a + 31 + \left(37 a + 18\right)\cdot 53 + \left(14 a + 46\right)\cdot 53^{2} + \left(23 a + 39\right)\cdot 53^{3} + \left(31 a + 49\right)\cdot 53^{4} + \left(33 a + 28\right)\cdot 53^{5} + \left(22 a + 48\right)\cdot 53^{6} + \left(36 a + 47\right)\cdot 53^{7} +O(53^{8})\)
$r_{ 2 }$ |
$=$ |
\( 50 a + 43 + \left(15 a + 4\right)\cdot 53 + \left(38 a + 15\right)\cdot 53^{2} + \left(29 a + 12\right)\cdot 53^{3} + \left(21 a + 46\right)\cdot 53^{4} + \left(19 a + 25\right)\cdot 53^{5} + \left(30 a + 52\right)\cdot 53^{6} + \left(16 a + 11\right)\cdot 53^{7} +O(53^{8})\)
| $r_{ 3 }$ |
$=$ |
\( 43 + 22\cdot 53 + 29\cdot 53^{2} + 17\cdot 53^{3} + 5\cdot 53^{4} + 37\cdot 53^{5} + 46\cdot 53^{6} + 31\cdot 53^{7} +O(53^{8})\)
| $r_{ 4 }$ |
$=$ |
\( 10 a + 33 + \left(14 a + 29\right)\cdot 53 + \left(14 a + 31\right)\cdot 53^{2} + \left(39 a + 34\right)\cdot 53^{3} + \left(39 a + 19\right)\cdot 53^{4} + \left(26 a + 19\right)\cdot 53^{5} + \left(10 a + 45\right)\cdot 53^{6} + \left(7 a + 43\right)\cdot 53^{7} +O(53^{8})\)
| $r_{ 5 }$ |
$=$ |
\( 50 a + 22 + \left(15 a + 34\right)\cdot 53 + \left(38 a + 6\right)\cdot 53^{2} + \left(29 a + 13\right)\cdot 53^{3} + \left(21 a + 3\right)\cdot 53^{4} + \left(19 a + 24\right)\cdot 53^{5} + \left(30 a + 4\right)\cdot 53^{6} + \left(16 a + 5\right)\cdot 53^{7} +O(53^{8})\)
| $r_{ 6 }$ |
$=$ |
\( 3 a + 10 + \left(37 a + 48\right)\cdot 53 + \left(14 a + 37\right)\cdot 53^{2} + \left(23 a + 40\right)\cdot 53^{3} + \left(31 a + 6\right)\cdot 53^{4} + \left(33 a + 27\right)\cdot 53^{5} + 22 a\cdot 53^{6} + \left(36 a + 41\right)\cdot 53^{7} +O(53^{8})\)
| $r_{ 7 }$ |
$=$ |
\( 10 + 30\cdot 53 + 23\cdot 53^{2} + 35\cdot 53^{3} + 47\cdot 53^{4} + 15\cdot 53^{5} + 6\cdot 53^{6} + 21\cdot 53^{7} +O(53^{8})\)
| $r_{ 8 }$ |
$=$ |
\( 43 a + 20 + \left(38 a + 23\right)\cdot 53 + \left(38 a + 21\right)\cdot 53^{2} + \left(13 a + 18\right)\cdot 53^{3} + \left(13 a + 33\right)\cdot 53^{4} + \left(26 a + 33\right)\cdot 53^{5} + \left(42 a + 7\right)\cdot 53^{6} + \left(45 a + 9\right)\cdot 53^{7} +O(53^{8})\)
| |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $-2$ |
$12$ | $2$ | $(1,5)(2,8)(4,6)$ | $0$ |
$8$ | $3$ | $(1,7,4)(3,8,5)$ | $-1$ |
$6$ | $4$ | $(1,6,5,2)(3,8,7,4)$ | $0$ |
$8$ | $6$ | $(1,8,7,5,4,3)(2,6)$ | $1$ |
$6$ | $8$ | $(1,2,7,4,5,6,3,8)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
$6$ | $8$ | $(1,6,7,8,5,2,3,4)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.