Basic invariants
| Dimension: | $2$ |
| Group: | $S_3\times C_3$ |
| Conductor: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Artin stem field: | Galois closure of 6.0.21168.1 |
| 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 3.1.588.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{5} + 4x^{4} - x^{3} - 2x^{2} + x + 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 21 a + 17 + \left(8 a + 17\right)\cdot 29 + \left(25 a + 14\right)\cdot 29^{2} + \left(24 a + 20\right)\cdot 29^{3} + \left(3 a + 3\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 12 a + 28 + \left(27 a + 15\right)\cdot 29 + \left(11 a + 12\right)\cdot 29^{2} + \left(20 a + 19\right)\cdot 29^{3} + \left(21 a + 22\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 8 a + 6 + \left(20 a + 11\right)\cdot 29 + \left(3 a + 16\right)\cdot 29^{2} + \left(4 a + 3\right)\cdot 29^{3} + \left(25 a + 27\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 17 a + 1 + \left(a + 25\right)\cdot 29 + \left(17 a + 15\right)\cdot 29^{2} + \left(8 a + 22\right)\cdot 29^{3} + \left(7 a + 23\right)\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 6 a + 4 + \left(2 a + 6\right)\cdot 29 + \left(a + 12\right)\cdot 29^{2} + \left(14 a + 19\right)\cdot 29^{3} + 4 a\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 23 a + 5 + \left(26 a + 11\right)\cdot 29 + \left(27 a + 15\right)\cdot 29^{2} + \left(14 a + 1\right)\cdot 29^{3} + \left(24 a + 9\right)\cdot 29^{4} +O(29^{5})\)
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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 | Complex conjugation |
| $1$ | $1$ | $()$ | $2$ | |
| $3$ | $2$ | $(1,3)(2,4)(5,6)$ | $0$ | ✓ |
| $1$ | $3$ | $(1,4,5)(2,6,3)$ | $-2 \zeta_{3} - 2$ | |
| $1$ | $3$ | $(1,5,4)(2,3,6)$ | $2 \zeta_{3}$ | |
| $2$ | $3$ | $(1,5,4)$ | $\zeta_{3} + 1$ | |
| $2$ | $3$ | $(1,4,5)$ | $-\zeta_{3}$ | |
| $2$ | $3$ | $(1,5,4)(2,6,3)$ | $-1$ | |
| $3$ | $6$ | $(1,6,4,3,5,2)$ | $0$ | |
| $3$ | $6$ | $(1,2,5,3,4,6)$ | $0$ |