Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(403\)\(\medspace = 13 \cdot 31 \) |
| Artin field: | Galois closure of 6.6.342896882653.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{403}(36,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 135x^{4} - 74x^{3} + 4478x^{2} + 9045x - 11493 \)
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The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$:
\( x^{2} + 33x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 a + 34 + \left(23 a + 32\right)\cdot 37 + \left(20 a + 36\right)\cdot 37^{2} + \left(31 a + 20\right)\cdot 37^{3} + \left(29 a + 21\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 33 a + 32 + \left(19 a + 17\right)\cdot 37 + \left(13 a + 35\right)\cdot 37^{2} + \left(28 a + 26\right)\cdot 37^{3} + \left(7 a + 24\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 35 a + 5 + \left(13 a + 12\right)\cdot 37 + \left(16 a + 22\right)\cdot 37^{2} + \left(5 a + 15\right)\cdot 37^{3} + \left(7 a + 35\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 18 a + 32 + \left(8 a + 20\right)\cdot 37 + \left(35 a + 36\right)\cdot 37^{2} + \left(36 a + 14\right)\cdot 37^{3} + \left(15 a + 24\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 19 a + 30 + \left(28 a + 36\right)\cdot 37 + \left(a + 20\right)\cdot 37^{2} + 16\cdot 37^{3} + \left(21 a + 14\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 4 a + 16 + \left(17 a + 27\right)\cdot 37 + \left(23 a + 32\right)\cdot 37^{2} + \left(8 a + 15\right)\cdot 37^{3} + \left(29 a + 27\right)\cdot 37^{4} +O(37^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,3)(2,6)(4,5)$ | $-1$ |
| $1$ | $3$ | $(1,2,4)(3,6,5)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,4,2)(3,5,6)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,6,4,3,2,5)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,5,2,3,4,6)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.