Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(273\)\(\medspace = 3 \cdot 7 \cdot 13 \) |
| Artin field: | Galois closure of 6.6.168488679177.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{273}(101,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 83x^{4} + 69x^{3} + 1527x^{2} + 153x - 27 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 28 a + 25 + \left(17 a + 23\right)\cdot 61 + \left(57 a + 56\right)\cdot 61^{2} + \left(58 a + 34\right)\cdot 61^{3} + \left(42 a + 59\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 25 a + 27 + \left(51 a + 52\right)\cdot 61 + \left(52 a + 31\right)\cdot 61^{2} + \left(48 a + 58\right)\cdot 61^{3} + \left(25 a + 11\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 4 a + 42 + \left(53 a + 13\right)\cdot 61 + \left(39 a + 50\right)\cdot 61^{2} + \left(52 a + 53\right)\cdot 61^{3} + \left(18 a + 25\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 33 a + 53 + \left(43 a + 12\right)\cdot 61 + \left(3 a + 35\right)\cdot 61^{2} + \left(2 a + 36\right)\cdot 61^{3} + \left(18 a + 43\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 57 a + 46 + \left(7 a + 1\right)\cdot 61 + \left(21 a + 37\right)\cdot 61^{2} + \left(8 a + 5\right)\cdot 61^{3} + \left(42 a + 53\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 36 a + 52 + \left(9 a + 17\right)\cdot 61 + \left(8 a + 33\right)\cdot 61^{2} + \left(12 a + 54\right)\cdot 61^{3} + \left(35 a + 49\right)\cdot 61^{4} +O(61^{5})\)
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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 | Complex conjugation |
| $1$ | $1$ | $()$ | $1$ | ✓ |
| $1$ | $2$ | $(1,4)(2,6)(3,5)$ | $-1$ | |
| $1$ | $3$ | $(1,2,5)(3,4,6)$ | $\zeta_{3}$ | |
| $1$ | $3$ | $(1,5,2)(3,6,4)$ | $-\zeta_{3} - 1$ | |
| $1$ | $6$ | $(1,3,2,4,5,6)$ | $\zeta_{3} + 1$ | |
| $1$ | $6$ | $(1,6,5,4,2,3)$ | $-\zeta_{3}$ |