Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(273\)\(\medspace = 3 \cdot 7 \cdot 13 \) |
| Artin field: | Galois closure of 6.6.12960667629.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{273}(152,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 62x^{4} - 36x^{3} + 813x^{2} + 1413x + 603 \)
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The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$:
\( x^{2} + 60x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 41 a + 49 + \left(39 a + 18\right)\cdot 61 + \left(27 a + 21\right)\cdot 61^{2} + \left(26 a + 36\right)\cdot 61^{3} + \left(30 a + 49\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 26 a + 31 + \left(33 a + 34\right)\cdot 61 + \left(35 a + 42\right)\cdot 61^{2} + \left(28 a + 2\right)\cdot 61^{3} + \left(18 a + 14\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 20 a + 29 + \left(21 a + 17\right)\cdot 61 + \left(33 a + 9\right)\cdot 61^{2} + \left(34 a + 35\right)\cdot 61^{3} + \left(30 a + 53\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 35 a + 57 + \left(27 a + 41\right)\cdot 61 + \left(25 a + 44\right)\cdot 61^{2} + \left(32 a + 56\right)\cdot 61^{3} + \left(42 a + 3\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 19 a + 30 + \left(30 a + 29\right)\cdot 61 + \left(40 a + 27\right)\cdot 61^{2} + \left(26 a + 2\right)\cdot 61^{3} + 27 a\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 42 a + 49 + \left(30 a + 40\right)\cdot 61 + \left(20 a + 37\right)\cdot 61^{2} + \left(34 a + 49\right)\cdot 61^{3} + 33 a\cdot 61^{4} +O(61^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,3)(2,4)(5,6)$ | $-1$ |
| $1$ | $3$ | $(1,5,4)(2,3,6)$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,4,5)(2,6,3)$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,2,5,3,4,6)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,6,4,3,5,2)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.