Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(665\)\(\medspace = 5 \cdot 7 \cdot 19 \) |
| Artin field: | Galois closure of 6.6.39112590125.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{665}(144,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 92x^{4} - 37x^{3} + 2004x^{2} + 3335x - 505 \)
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The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$:
\( x^{2} + 49x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 43 a + 48 + \left(9 a + 39\right)\cdot 53 + \left(17 a + 18\right)\cdot 53^{2} + \left(36 a + 40\right)\cdot 53^{3} + \left(35 a + 25\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 10 a + 40 + \left(43 a + 22\right)\cdot 53 + \left(35 a + 36\right)\cdot 53^{2} + \left(16 a + 43\right)\cdot 53^{3} + \left(17 a + 6\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 43 a + 27 + \left(9 a + 26\right)\cdot 53 + \left(17 a + 30\right)\cdot 53^{2} + \left(36 a + 21\right)\cdot 53^{3} + \left(35 a + 6\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 43 a + 12 + \left(9 a + 45\right)\cdot 53 + \left(17 a + 47\right)\cdot 53^{2} + \left(36 a + 10\right)\cdot 53^{3} + \left(35 a + 20\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 10 a + 8 + \left(43 a + 36\right)\cdot 53 + \left(35 a + 24\right)\cdot 53^{2} + \left(16 a + 9\right)\cdot 53^{3} + \left(17 a + 26\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 10 a + 25 + \left(43 a + 41\right)\cdot 53 + 35 a\cdot 53^{2} + \left(16 a + 33\right)\cdot 53^{3} + \left(17 a + 20\right)\cdot 53^{4} +O(53^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,5)(2,3)(4,6)$ | $-1$ |
| $1$ | $3$ | $(1,3,4)(2,6,5)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,4,3)(2,5,6)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,6,3,5,4,2)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,2,4,5,3,6)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.