Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(3564\)\(\medspace = 2^{2} \cdot 3^{4} \cdot 11 \) |
| Artin stem field: | Galois closure of 6.0.1676676672.9 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $S_3\times C_3$ |
| Parity: | odd |
| Determinant: | 1.396.6t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.1188.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + 12x^{4} - 12x^{3} + 36x^{2} - 72x + 168 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$:
\( x^{2} + 49x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 50 a + 31 + \left(40 a + 7\right)\cdot 53 + \left(47 a + 47\right)\cdot 53^{2} + \left(29 a + 52\right)\cdot 53^{3} + \left(50 a + 51\right)\cdot 53^{4} + \left(a + 1\right)\cdot 53^{5} +O(53^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 52 a + 20 + \left(37 a + 22\right)\cdot 53 + \left(40 a + 47\right)\cdot 53^{2} + \left(18 a + 38\right)\cdot 53^{3} + \left(28 a + 4\right)\cdot 53^{4} + \left(12 a + 7\right)\cdot 53^{5} +O(53^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 49 a + 18 + \left(25 a + 21\right)\cdot 53 + \left(35 a + 1\right)\cdot 53^{2} + \left(48 a + 14\right)\cdot 53^{3} + \left(25 a + 47\right)\cdot 53^{4} + \left(14 a + 11\right)\cdot 53^{5} +O(53^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( a + 16 + \left(15 a + 16\right)\cdot 53 + \left(12 a + 13\right)\cdot 53^{2} + \left(34 a + 20\right)\cdot 53^{3} + \left(24 a + 46\right)\cdot 53^{4} + \left(40 a + 28\right)\cdot 53^{5} +O(53^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 4 a + 2 + \left(27 a + 23\right)\cdot 53 + \left(17 a + 11\right)\cdot 53^{2} + \left(4 a + 14\right)\cdot 53^{3} + \left(27 a + 49\right)\cdot 53^{4} + \left(38 a + 43\right)\cdot 53^{5} +O(53^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 3 a + 19 + \left(12 a + 15\right)\cdot 53 + \left(5 a + 38\right)\cdot 53^{2} + \left(23 a + 18\right)\cdot 53^{3} + \left(2 a + 12\right)\cdot 53^{4} + \left(51 a + 12\right)\cdot 53^{5} +O(53^{6})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 6 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 6 }$ | Character value |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,3)(2,6)(4,5)$ | $0$ |
| $1$ | $3$ | $(1,5,2)(3,4,6)$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,2,5)(3,6,4)$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(1,2,5)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,5,2)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,5,2)(3,6,4)$ | $-1$ |
| $3$ | $6$ | $(1,6,5,3,2,4)$ | $0$ |
| $3$ | $6$ | $(1,4,2,3,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.