Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(247\)\(\medspace = 13 \cdot 19 \) |
| Artin field: | Galois closure of 6.0.5439989503.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{247}(103,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 59x^{4} + 65x^{3} + 341x^{2} + 337x + 1147 \)
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The roots of $f$ are computed in an extension of $\Q_{ 83 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 83 }$:
\( x^{2} + 82x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 44 a + 29 + \left(13 a + 17\right)\cdot 83 + \left(37 a + 32\right)\cdot 83^{2} + \left(29 a + 27\right)\cdot 83^{3} + \left(5 a + 43\right)\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( a + 12 + 74 a\cdot 83 + \left(75 a + 20\right)\cdot 83^{2} + \left(45 a + 16\right)\cdot 83^{3} + \left(23 a + 31\right)\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 82 a + 13 + \left(8 a + 73\right)\cdot 83 + \left(7 a + 21\right)\cdot 83^{2} + \left(37 a + 69\right)\cdot 83^{3} + \left(59 a + 8\right)\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 60 a + 73 + \left(50 a + 48\right)\cdot 83 + \left(31 a + 27\right)\cdot 83^{2} + \left(78 a + 76\right)\cdot 83^{3} + \left(27 a + 56\right)\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 39 a + 73 + \left(69 a + 69\right)\cdot 83 + \left(45 a + 55\right)\cdot 83^{2} + \left(53 a + 19\right)\cdot 83^{3} + \left(77 a + 19\right)\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 23 a + 50 + \left(32 a + 39\right)\cdot 83 + \left(51 a + 8\right)\cdot 83^{2} + \left(4 a + 40\right)\cdot 83^{3} + \left(55 a + 6\right)\cdot 83^{4} +O(83^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ | |
| $1$ | $2$ | $(1,5)(2,3)(4,6)$ | $-1$ | ✓ |
| $1$ | $3$ | $(1,6,2)(3,5,4)$ | $\zeta_{3}$ | |
| $1$ | $3$ | $(1,2,6)(3,4,5)$ | $-\zeta_{3} - 1$ | |
| $1$ | $6$ | $(1,4,2,5,6,3)$ | $-\zeta_{3}$ | |
| $1$ | $6$ | $(1,3,6,5,2,4)$ | $\zeta_{3} + 1$ |