Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(39\)\(\medspace = 3 \cdot 13 \) |
| Artin field: | Galois closure of 6.0.10024911.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{39}(23,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 8x^{4} - 9x^{3} + 6x^{2} + 10x + 25 \)
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The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 29 a + 5 + \left(7 a + 9\right)\cdot 31 + \left(11 a + 11\right)\cdot 31^{2} + \left(7 a + 13\right)\cdot 31^{3} + \left(21 a + 27\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 12 a + 30 + \left(24 a + 6\right)\cdot 31 + \left(28 a + 11\right)\cdot 31^{2} + \left(4 a + 20\right)\cdot 31^{3} + \left(18 a + 29\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 2 a + 1 + \left(23 a + 27\right)\cdot 31 + \left(19 a + 25\right)\cdot 31^{2} + \left(23 a + 16\right)\cdot 31^{3} + 9 a\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 16 a + 17 + \left(20 a + 21\right)\cdot 31 + \left(20 a + 20\right)\cdot 31^{2} + \left(4 a + 10\right)\cdot 31^{3} + \left(17 a + 3\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 19 a + 23 + \left(6 a + 12\right)\cdot 31 + \left(2 a + 13\right)\cdot 31^{2} + \left(26 a + 1\right)\cdot 31^{3} + \left(12 a + 30\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 15 a + 18 + \left(10 a + 15\right)\cdot 31 + \left(10 a + 10\right)\cdot 31^{2} + \left(26 a + 30\right)\cdot 31^{3} + \left(13 a + 1\right)\cdot 31^{4} +O(31^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,3)(2,5)(4,6)$ | $-1$ |
| $1$ | $3$ | $(1,5,6)(2,4,3)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,6,5)(2,3,4)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,4,5,3,6,2)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,2,6,3,5,4)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.