Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(1267\)\(\medspace = 7 \cdot 181 \) |
| Artin field: | Galois closure of 6.6.14237308141.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{1267}(723,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 140x^{4} + 93x^{3} + 6125x^{2} - 2116x - 83161 \)
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The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$:
\( x^{2} + 38x + 6 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 31 a + 9 + \left(a + 2\right)\cdot 41 + \left(29 a + 36\right)\cdot 41^{2} + \left(34 a + 9\right)\cdot 41^{3} + \left(15 a + 23\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 31 a + 32 + \left(a + 39\right)\cdot 41 + \left(29 a + 12\right)\cdot 41^{2} + \left(34 a + 35\right)\cdot 41^{3} + \left(15 a + 3\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 31 a + 25 + \left(a + 37\right)\cdot 41 + \left(29 a + 27\right)\cdot 41^{2} + \left(34 a + 26\right)\cdot 41^{3} + \left(15 a + 35\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 10 a + 2 + \left(39 a + 14\right)\cdot 41 + \left(11 a + 16\right)\cdot 41^{2} + \left(6 a + 28\right)\cdot 41^{3} + \left(25 a + 16\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 10 a + 36 + \left(39 a + 11\right)\cdot 41 + \left(11 a + 31\right)\cdot 41^{2} + \left(6 a + 19\right)\cdot 41^{3} + \left(25 a + 7\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 10 a + 20 + \left(39 a + 17\right)\cdot 41 + \left(11 a + 39\right)\cdot 41^{2} + \left(6 a + 2\right)\cdot 41^{3} + \left(25 a + 36\right)\cdot 41^{4} +O(41^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,6)(2,4)(3,5)$ | $-1$ |
| $1$ | $3$ | $(1,3,2)(4,6,5)$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,2,3)(4,5,6)$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,5,2,6,3,4)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,4,3,6,2,5)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.