Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(756\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 7 \) |
| Artin stem field: | Galois closure of 6.0.1714608.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $S_3\times C_3$ |
| Parity: | odd |
| Determinant: | 1.21.6t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.5292.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{5} - 6x^{4} + 13x^{3} + 24x^{2} + 9x + 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( a + 14 + \left(19 a + 17\right)\cdot 29 + \left(22 a + 14\right)\cdot 29^{2} + \left(13 a + 26\right)\cdot 29^{3} + \left(13 a + 14\right)\cdot 29^{4} + \left(3 a + 1\right)\cdot 29^{5} + \left(10 a + 20\right)\cdot 29^{6} +O(29^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 12 a + 6 + \left(24 a + 21\right)\cdot 29 + \left(12 a + 8\right)\cdot 29^{2} + \left(14 a + 1\right)\cdot 29^{3} + \left(11 a + 18\right)\cdot 29^{4} + \left(5 a + 10\right)\cdot 29^{5} + \left(8 a + 23\right)\cdot 29^{6} +O(29^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 28 a + 19 + \left(9 a + 24\right)\cdot 29 + \left(6 a + 21\right)\cdot 29^{2} + \left(15 a + 14\right)\cdot 29^{3} + \left(15 a + 10\right)\cdot 29^{4} + \left(25 a + 5\right)\cdot 29^{5} + \left(18 a + 9\right)\cdot 29^{6} +O(29^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 4 a + 26 + \left(7 a + 2\right)\cdot 29 + \left(24 a + 12\right)\cdot 29^{2} + \left(17 a + 17\right)\cdot 29^{3} + \left(17 a + 28\right)\cdot 29^{4} + \left(28 a + 16\right)\cdot 29^{5} + \left(3 a + 6\right)\cdot 29^{6} +O(29^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 17 a + 8 + \left(4 a + 15\right)\cdot 29 + \left(16 a + 19\right)\cdot 29^{2} + \left(14 a + 2\right)\cdot 29^{3} + \left(17 a + 3\right)\cdot 29^{4} + \left(23 a + 26\right)\cdot 29^{5} + 20 a\cdot 29^{6} +O(29^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 25 a + 17 + \left(21 a + 5\right)\cdot 29 + \left(4 a + 10\right)\cdot 29^{2} + \left(11 a + 24\right)\cdot 29^{3} + \left(11 a + 11\right)\cdot 29^{4} + 26\cdot 29^{5} + \left(25 a + 26\right)\cdot 29^{6} +O(29^{7})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,3)(2,5)(4,6)$ | $0$ |
| $1$ | $3$ | $(1,5,6)(2,4,3)$ | $2 \zeta_{3}$ |
| $1$ | $3$ | $(1,6,5)(2,3,4)$ | $-2 \zeta_{3} - 2$ |
| $2$ | $3$ | $(1,5,6)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,6,5)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,6,5)(2,4,3)$ | $-1$ |
| $3$ | $6$ | $(1,3,5,2,6,4)$ | $0$ |
| $3$ | $6$ | $(1,4,6,2,5,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.