Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(473\)\(\medspace = 11 \cdot 43 \) |
| Artin field: | Galois closure of 6.0.4550424131.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{473}(307,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 20x^{4} + 39x^{3} + 212x^{2} - 649x + 1067 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 30 a + 40 + \left(15 a + 21\right)\cdot 41 + \left(31 a + 15\right)\cdot 41^{2} + \left(20 a + 1\right)\cdot 41^{3} + \left(8 a + 2\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 30 a + 1 + \left(15 a + 6\right)\cdot 41 + \left(31 a + 16\right)\cdot 41^{2} + \left(20 a + 12\right)\cdot 41^{3} + \left(8 a + 24\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 30 a + 9 + \left(15 a + 28\right)\cdot 41 + \left(31 a + 14\right)\cdot 41^{2} + \left(20 a + 1\right)\cdot 41^{3} + \left(8 a + 28\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 11 a + 17 + \left(25 a + 4\right)\cdot 41 + \left(9 a + 11\right)\cdot 41^{2} + \left(20 a + 32\right)\cdot 41^{3} + \left(32 a + 32\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 11 a + 7 + \left(25 a + 39\right)\cdot 41 + \left(9 a + 11\right)\cdot 41^{2} + \left(20 a + 32\right)\cdot 41^{3} + \left(32 a + 6\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 11 a + 9 + \left(25 a + 23\right)\cdot 41 + \left(9 a + 12\right)\cdot 41^{2} + \left(20 a + 2\right)\cdot 41^{3} + \left(32 a + 29\right)\cdot 41^{4} +O(41^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ | |
| $1$ | $2$ | $(1,5)(2,6)(3,4)$ | $-1$ | ✓ |
| $1$ | $3$ | $(1,2,3)(4,5,6)$ | $-\zeta_{3} - 1$ | |
| $1$ | $3$ | $(1,3,2)(4,6,5)$ | $\zeta_{3}$ | |
| $1$ | $6$ | $(1,6,3,5,2,4)$ | $\zeta_{3} + 1$ | |
| $1$ | $6$ | $(1,4,2,5,3,6)$ | $-\zeta_{3}$ |