Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(312\)\(\medspace = 2^{3} \cdot 3 \cdot 13 \) |
| Artin field: | Galois closure of 6.0.394827264.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{312}(269,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 11x^{4} - 18x^{3} + 138x^{2} + 20x + 625 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 5 a + 9 + \left(24 a + 38\right)\cdot 47 + \left(8 a + 19\right)\cdot 47^{2} + \left(33 a + 26\right)\cdot 47^{3} + \left(2 a + 25\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 5 a + 20 + \left(24 a + 1\right)\cdot 47 + \left(8 a + 7\right)\cdot 47^{2} + \left(33 a + 39\right)\cdot 47^{3} + \left(2 a + 41\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 5 a + 4 + \left(24 a + 13\right)\cdot 47 + \left(8 a + 7\right)\cdot 47^{2} + \left(33 a + 12\right)\cdot 47^{3} + \left(2 a + 21\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 42 a + 30 + \left(22 a + 44\right)\cdot 47 + \left(38 a + 46\right)\cdot 47^{2} + \left(13 a + 2\right)\cdot 47^{3} + \left(44 a + 14\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 42 a + 14 + \left(22 a + 9\right)\cdot 47 + 38 a\cdot 47^{2} + \left(13 a + 23\right)\cdot 47^{3} + \left(44 a + 40\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 42 a + 19 + \left(22 a + 34\right)\cdot 47 + \left(38 a + 12\right)\cdot 47^{2} + \left(13 a + 37\right)\cdot 47^{3} + \left(44 a + 44\right)\cdot 47^{4} +O(47^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ | |
| $1$ | $2$ | $(1,6)(2,4)(3,5)$ | $-1$ | ✓ |
| $1$ | $3$ | $(1,3,2)(4,6,5)$ | $\zeta_{3}$ | |
| $1$ | $3$ | $(1,2,3)(4,5,6)$ | $-\zeta_{3} - 1$ | |
| $1$ | $6$ | $(1,5,2,6,3,4)$ | $-\zeta_{3}$ | |
| $1$ | $6$ | $(1,4,3,6,2,5)$ | $\zeta_{3} + 1$ |