Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(315\)\(\medspace = 3^{2} \cdot 5 \cdot 7 \) |
| Artin field: | Galois closure of 6.6.1969120125.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{315}(184,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{5} - 42x^{4} + 33x^{3} + 462x^{2} + 501x - 71 \)
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The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$:
\( x^{2} + 45x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 19 a + 28 + \left(21 a + 31\right)\cdot 47 + \left(29 a + 11\right)\cdot 47^{2} + \left(9 a + 40\right)\cdot 47^{3} + \left(30 a + 10\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 28 a + 32 + \left(25 a + 2\right)\cdot 47 + \left(17 a + 43\right)\cdot 47^{2} + \left(37 a + 30\right)\cdot 47^{3} + \left(16 a + 42\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 19 a + 40 + 21 a\cdot 47 + \left(29 a + 44\right)\cdot 47^{2} + \left(9 a + 27\right)\cdot 47^{3} + \left(30 a + 38\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 19 a + 41 + \left(21 a + 25\right)\cdot 47 + \left(29 a + 5\right)\cdot 47^{2} + \left(9 a + 41\right)\cdot 47^{3} + \left(30 a + 38\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 28 a + 19 + \left(25 a + 8\right)\cdot 47 + \left(17 a + 2\right)\cdot 47^{2} + \left(37 a + 30\right)\cdot 47^{3} + \left(16 a + 14\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 28 a + 31 + \left(25 a + 24\right)\cdot 47 + \left(17 a + 34\right)\cdot 47^{2} + \left(37 a + 17\right)\cdot 47^{3} + \left(16 a + 42\right)\cdot 47^{4} +O(47^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ | ✓ |
| $1$ | $2$ | $(1,5)(2,4)(3,6)$ | $-1$ | |
| $1$ | $3$ | $(1,4,3)(2,6,5)$ | $-\zeta_{3} - 1$ | |
| $1$ | $3$ | $(1,3,4)(2,5,6)$ | $\zeta_{3}$ | |
| $1$ | $6$ | $(1,6,4,5,3,2)$ | $-\zeta_{3}$ | |
| $1$ | $6$ | $(1,2,3,5,4,6)$ | $\zeta_{3} + 1$ |