Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(1143\)\(\medspace = 3^{2} \cdot 127 \) |
| Artin field: | Galois closure of 6.0.13439440863.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{1143}(1015,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{5} + 93x^{4} - 179x^{3} + 3165x^{2} - 3273x + 39203 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 2 a + 5 + \left(39 a + 34\right)\cdot 53 + \left(14 a + 30\right)\cdot 53^{2} + \left(2 a + 2\right)\cdot 53^{3} + \left(47 a + 43\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 2 a + 27 + \left(39 a + 22\right)\cdot 53 + \left(14 a + 29\right)\cdot 53^{2} + \left(2 a + 46\right)\cdot 53^{3} + \left(47 a + 40\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 2 a + 37 + \left(39 a + 3\right)\cdot 53 + \left(14 a + 16\right)\cdot 53^{2} + \left(2 a + 12\right)\cdot 53^{3} + \left(47 a + 8\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 51 a + 35 + \left(13 a + 17\right)\cdot 53 + \left(38 a + 49\right)\cdot 53^{2} + \left(50 a + 40\right)\cdot 53^{3} + \left(5 a + 14\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 51 a + 13 + \left(13 a + 29\right)\cdot 53 + \left(38 a + 50\right)\cdot 53^{2} + \left(50 a + 49\right)\cdot 53^{3} + \left(5 a + 16\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 51 a + 45 + \left(13 a + 51\right)\cdot 53 + \left(38 a + 35\right)\cdot 53^{2} + \left(50 a + 6\right)\cdot 53^{3} + \left(5 a + 35\right)\cdot 53^{4} +O(53^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,5)(2,4)(3,6)$ | $-1$ |
| $1$ | $3$ | $(1,2,3)(4,6,5)$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,3,2)(4,5,6)$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,4,3,5,2,6)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,6,2,5,3,4)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.