Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(273\)\(\medspace = 3 \cdot 7 \cdot 13 \) |
| Artin field: | Galois closure of 6.6.3438544473.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{273}(62,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 70x^{4} + 69x^{3} + 786x^{2} - 68x - 1301 \)
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The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$:
\( x^{2} + 45x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 26 a + 2 + \left(33 a + 32\right)\cdot 47 + \left(5 a + 24\right)\cdot 47^{2} + \left(44 a + 14\right)\cdot 47^{3} + \left(13 a + 35\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 21 a + 7 + \left(13 a + 26\right)\cdot 47 + \left(41 a + 2\right)\cdot 47^{2} + \left(2 a + 3\right)\cdot 47^{3} + \left(33 a + 19\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 46 a + 8 + \left(23 a + 17\right)\cdot 47 + \left(41 a + 37\right)\cdot 47^{2} + \left(21 a + 14\right)\cdot 47^{3} + \left(13 a + 3\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( a + 6 + \left(23 a + 19\right)\cdot 47 + \left(5 a + 2\right)\cdot 47^{2} + \left(25 a + 17\right)\cdot 47^{3} + \left(33 a + 8\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 14 a + 22 + \left(29 a + 24\right)\cdot 47 + \left(4 a + 23\right)\cdot 47^{2} + \left(21 a + 3\right)\cdot 47^{3} + \left(9 a + 15\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 33 a + 3 + \left(17 a + 22\right)\cdot 47 + \left(42 a + 3\right)\cdot 47^{2} + \left(25 a + 41\right)\cdot 47^{3} + \left(37 a + 12\right)\cdot 47^{4} +O(47^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ | ✓ |
| $1$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ | |
| $1$ | $3$ | $(1,3,6)(2,4,5)$ | $\zeta_{3}$ | |
| $1$ | $3$ | $(1,6,3)(2,5,4)$ | $-\zeta_{3} - 1$ | |
| $1$ | $6$ | $(1,4,6,2,3,5)$ | $-\zeta_{3}$ | |
| $1$ | $6$ | $(1,5,3,2,6,4)$ | $\zeta_{3} + 1$ |