Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(91\)\(\medspace = 7 \cdot 13 \) |
| Artin field: | Galois closure of 6.0.480024727.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{91}(3,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + x^{4} + 13x^{3} + 71x^{2} + 419x + 827 \)
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The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$:
\( x^{2} + 60x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 10 a + 39 + 24 a\cdot 61 + \left(53 a + 29\right)\cdot 61^{2} + \left(59 a + 26\right)\cdot 61^{3} + \left(12 a + 32\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 49 a + 15 + \left(4 a + 57\right)\cdot 61 + \left(45 a + 42\right)\cdot 61^{2} + \left(47 a + 24\right)\cdot 61^{3} + \left(26 a + 41\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 12 a + 3 + \left(56 a + 13\right)\cdot 61 + \left(15 a + 22\right)\cdot 61^{2} + \left(13 a + 27\right)\cdot 61^{3} + \left(34 a + 20\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 58 a + 10 + \left(2 a + 15\right)\cdot 61 + \left(56 a + 19\right)\cdot 61^{2} + \left(50 a + 38\right)\cdot 61^{3} + \left(43 a + 24\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 51 a + 49 + \left(36 a + 14\right)\cdot 61 + \left(7 a + 58\right)\cdot 61^{2} + \left(a + 32\right)\cdot 61^{3} + \left(48 a + 46\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 3 a + 7 + \left(58 a + 21\right)\cdot 61 + \left(4 a + 11\right)\cdot 61^{2} + \left(10 a + 33\right)\cdot 61^{3} + \left(17 a + 17\right)\cdot 61^{4} +O(61^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,5)(2,3)(4,6)$ | $-1$ |
| $1$ | $3$ | $(1,6,2)(3,5,4)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,2,6)(3,4,5)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,4,2,5,6,3)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,3,6,5,2,4)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.