Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(1368\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 19 \) |
| Artin stem field: | Galois closure of 6.0.14971392.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $S_3\times C_3$ |
| Parity: | odd |
| Determinant: | 1.1368.6t1.b.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.233928.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 11x^{4} - 10x^{3} + 12x^{2} + 12x + 3 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 9 a + 3 + \left(27 a + 15\right)\cdot 29 + \left(11 a + 7\right)\cdot 29^{2} + \left(26 a + 13\right)\cdot 29^{3} + \left(7 a + 11\right)\cdot 29^{4} + \left(20 a + 7\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 23 a + 25 + \left(27 a + 22\right)\cdot 29 + \left(10 a + 22\right)\cdot 29^{2} + \left(16 a + 25\right)\cdot 29^{3} + \left(19 a + 20\right)\cdot 29^{4} + \left(12 a + 27\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 6 a + 24 + \left(a + 22\right)\cdot 29 + \left(18 a + 20\right)\cdot 29^{2} + \left(12 a + 9\right)\cdot 29^{3} + \left(9 a + 15\right)\cdot 29^{4} + \left(16 a + 13\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 21 a + \left(14 a + 17\right)\cdot 29 + \left(26 a + 11\right)\cdot 29^{2} + \left(12 a + 20\right)\cdot 29^{3} + \left(14 a + 6\right)\cdot 29^{4} + \left(22 a + 21\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 8 a + 18 + \left(14 a + 11\right)\cdot 29 + \left(2 a + 13\right)\cdot 29^{2} + 16 a\cdot 29^{3} + \left(14 a + 8\right)\cdot 29^{4} + \left(6 a + 3\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 20 a + 19 + \left(a + 26\right)\cdot 29 + \left(17 a + 10\right)\cdot 29^{2} + \left(2 a + 17\right)\cdot 29^{3} + \left(21 a + 24\right)\cdot 29^{4} + \left(8 a + 13\right)\cdot 29^{5} +O(29^{6})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,5)(2,4)(3,6)$ | $0$ |
| $1$ | $3$ | $(1,3,4)(2,5,6)$ | $2 \zeta_{3}$ |
| $1$ | $3$ | $(1,4,3)(2,6,5)$ | $-2 \zeta_{3} - 2$ |
| $2$ | $3$ | $(1,3,4)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,4,3)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,4,3)(2,5,6)$ | $-1$ |
| $3$ | $6$ | $(1,2,3,5,4,6)$ | $0$ |
| $3$ | $6$ | $(1,6,4,5,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.