Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(65\)\(\medspace = 5 \cdot 13 \) |
| Artin field: | Galois closure of 6.6.46411625.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{65}(4,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 18x^{4} + 17x^{3} + 58x^{2} - 16x - 1 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 20 a + 13 + \left(14 a + 29\right)\cdot 31 + \left(30 a + 7\right)\cdot 31^{2} + \left(7 a + 12\right)\cdot 31^{3} + 6 a\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 15 a + 27 + \left(14 a + 2\right)\cdot 31 + \left(27 a + 23\right)\cdot 31^{2} + \left(2 a + 21\right)\cdot 31^{3} + \left(25 a + 21\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 11 a + 22 + \left(16 a + 7\right)\cdot 31 + 23\cdot 31^{2} + \left(23 a + 28\right)\cdot 31^{3} + \left(24 a + 4\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 16 a + 26 + \left(16 a + 16\right)\cdot 31 + \left(3 a + 1\right)\cdot 31^{2} + 28 a\cdot 31^{3} + \left(5 a + 7\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 18 a + 16 + \left(23 a + 3\right)\cdot 31 + \left(23 a + 22\right)\cdot 31^{2} + \left(11 a + 30\right)\cdot 31^{3} + \left(9 a + 25\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 13 a + 21 + \left(7 a + 1\right)\cdot 31 + \left(7 a + 15\right)\cdot 31^{2} + \left(19 a + 30\right)\cdot 31^{3} + \left(21 a + 1\right)\cdot 31^{4} +O(31^{5})\)
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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 | Complex conjugation |
| $1$ | $1$ | $()$ | $1$ | ✓ |
| $1$ | $2$ | $(1,3)(2,4)(5,6)$ | $-1$ | |
| $1$ | $3$ | $(1,2,5)(3,4,6)$ | $\zeta_{3}$ | |
| $1$ | $3$ | $(1,5,2)(3,6,4)$ | $-\zeta_{3} - 1$ | |
| $1$ | $6$ | $(1,4,5,3,2,6)$ | $-\zeta_{3}$ | |
| $1$ | $6$ | $(1,6,2,3,5,4)$ | $\zeta_{3} + 1$ |