Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(536\)\(\medspace = 2^{3} \cdot 67 \) |
| Artin stem field: | Galois closure of 6.0.2298368.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $S_3\times C_3$ |
| Parity: | odd |
| Determinant: | 1.536.6t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.35912.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 4x^{4} - 2x^{3} + 11x^{2} - 6x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$:
\( x^{2} + 49x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 a + 27 + \left(16 a + 6\right)\cdot 53 + \left(29 a + 20\right)\cdot 53^{2} + \left(42 a + 3\right)\cdot 53^{3} + \left(32 a + 14\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 52 a + 26 + \left(7 a + 51\right)\cdot 53 + \left(19 a + 6\right)\cdot 53^{2} + \left(34 a + 28\right)\cdot 53^{3} + \left(37 a + 20\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 18 a + 7 + \left(23 a + 44\right)\cdot 53 + \left(36 a + 38\right)\cdot 53^{2} + \left(19 a + 29\right)\cdot 53^{3} + \left(32 a + 46\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 47 a + 51 + \left(36 a + 11\right)\cdot 53 + \left(23 a + 15\right)\cdot 53^{2} + \left(10 a + 38\right)\cdot 53^{3} + \left(20 a + 49\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 35 a + 26 + \left(29 a + 13\right)\cdot 53 + \left(16 a + 2\right)\cdot 53^{2} + \left(33 a + 19\right)\cdot 53^{3} + \left(20 a + 50\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( a + 22 + \left(45 a + 31\right)\cdot 53 + \left(33 a + 22\right)\cdot 53^{2} + \left(18 a + 40\right)\cdot 53^{3} + \left(15 a + 30\right)\cdot 53^{4} +O(53^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,5)(2,4)(3,6)$ | $0$ |
| $1$ | $3$ | $(1,2,3)(4,6,5)$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,3,2)(4,5,6)$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(4,6,5)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(4,5,6)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $3$ | $6$ | $(1,4,3,5,2,6)$ | $0$ |
| $3$ | $6$ | $(1,6,2,5,3,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.