Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(760\)\(\medspace = 2^{3} \cdot 5 \cdot 19 \) |
| Artin field: | Galois closure of 6.0.8340544000.3 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{760}(619,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 19x^{4} - 14x^{3} + 342x^{2} - 584x + 2849 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 6 a + 29 + \left(a + 11\right)\cdot 31 + \left(a + 27\right)\cdot 31^{2} + \left(11 a + 20\right)\cdot 31^{3} + \left(21 a + 19\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 25 a + 22 + 29 a\cdot 31 + \left(29 a + 22\right)\cdot 31^{2} + \left(19 a + 11\right)\cdot 31^{3} + \left(9 a + 9\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 6 a + 6 + \left(a + 20\right)\cdot 31 + \left(a + 27\right)\cdot 31^{2} + \left(11 a + 18\right)\cdot 31^{3} + \left(21 a + 1\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 6 a + 10 + \left(a + 4\right)\cdot 31 + \left(a + 21\right)\cdot 31^{2} + \left(11 a + 21\right)\cdot 31^{3} + \left(21 a + 8\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 25 a + 10 + \left(29 a + 8\right)\cdot 31 + \left(29 a + 28\right)\cdot 31^{2} + \left(19 a + 10\right)\cdot 31^{3} + \left(9 a + 20\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 25 a + 18 + \left(29 a + 16\right)\cdot 31 + \left(29 a + 28\right)\cdot 31^{2} + \left(19 a + 8\right)\cdot 31^{3} + \left(9 a + 2\right)\cdot 31^{4} +O(31^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,5)(2,4)(3,6)$ | $-1$ |
| $1$ | $3$ | $(1,4,3)(2,6,5)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,3,4)(2,5,6)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,6,4,5,3,2)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,2,3,5,4,6)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.