Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(231\)\(\medspace = 3 \cdot 7 \cdot 11 \) |
| Artin field: | Galois closure of 6.0.603993159.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{231}(131,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 57x^{4} - 57x^{3} + 953x^{2} - 953x + 4537 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 13 a + 27 + \left(12 a + 16\right)\cdot 41 + \left(5 a + 24\right)\cdot 41^{2} + \left(35 a + 30\right)\cdot 41^{3} + \left(12 a + 38\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 20 a + 13 + \left(15 a + 14\right)\cdot 41 + \left(29 a + 38\right)\cdot 41^{2} + \left(34 a + 38\right)\cdot 41^{3} + \left(14 a + 30\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 28 a + 25 + \left(28 a + 40\right)\cdot 41 + \left(35 a + 27\right)\cdot 41^{2} + \left(5 a + 7\right)\cdot 41^{3} + \left(28 a + 1\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 39 a + 37 + \left(13 a + 3\right)\cdot 41 + \left(34 a + 39\right)\cdot 41^{2} + \left(15 a + 20\right)\cdot 41^{3} + \left(21 a + 22\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 21 a + 32 + \left(25 a + 40\right)\cdot 41 + \left(11 a + 28\right)\cdot 41^{2} + \left(6 a + 31\right)\cdot 41^{3} + \left(26 a + 40\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 2 a + 31 + \left(27 a + 6\right)\cdot 41 + \left(6 a + 5\right)\cdot 41^{2} + \left(25 a + 34\right)\cdot 41^{3} + \left(19 a + 29\right)\cdot 41^{4} +O(41^{5})\)
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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 | Complex conjugation |
| $1$ | $1$ | $()$ | $1$ | |
| $1$ | $2$ | $(1,3)(2,5)(4,6)$ | $-1$ | ✓ |
| $1$ | $3$ | $(1,5,4)(2,6,3)$ | $-\zeta_{3} - 1$ | |
| $1$ | $3$ | $(1,4,5)(2,3,6)$ | $\zeta_{3}$ | |
| $1$ | $6$ | $(1,6,5,3,4,2)$ | $-\zeta_{3}$ | |
| $1$ | $6$ | $(1,2,4,3,5,6)$ | $\zeta_{3} + 1$ |