Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(728\)\(\medspace = 2^{3} \cdot 7 \cdot 13 \) |
| Artin field: | Galois closure of 6.0.35110380032.4 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{728}(627,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} - 53x^{4} - 2x^{3} + 970x^{2} + 2056x + 2673 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 55 a + 30 + \left(58 a + 13\right)\cdot 61 + \left(46 a + 32\right)\cdot 61^{2} + \left(60 a + 52\right)\cdot 61^{3} + \left(38 a + 28\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 6 a + 15 + \left(2 a + 11\right)\cdot 61 + \left(14 a + 59\right)\cdot 61^{2} + 58\cdot 61^{3} + \left(22 a + 50\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 6 a + 24 + \left(2 a + 17\right)\cdot 61 + \left(14 a + 20\right)\cdot 61^{2} + 5\cdot 61^{3} + \left(22 a + 7\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 6 a + 14 + \left(2 a + 38\right)\cdot 61 + \left(14 a + 24\right)\cdot 61^{2} + 17\cdot 61^{3} + \left(22 a + 31\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 55 a + 21 + \left(58 a + 7\right)\cdot 61 + \left(46 a + 10\right)\cdot 61^{2} + \left(60 a + 45\right)\cdot 61^{3} + \left(38 a + 11\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 55 a + 20 + \left(58 a + 34\right)\cdot 61 + \left(46 a + 36\right)\cdot 61^{2} + \left(60 a + 3\right)\cdot 61^{3} + \left(38 a + 53\right)\cdot 61^{4} +O(61^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ | |
| $1$ | $2$ | $(1,3)(2,5)(4,6)$ | $-1$ | ✓ |
| $1$ | $3$ | $(1,5,6)(2,4,3)$ | $-\zeta_{3} - 1$ | |
| $1$ | $3$ | $(1,6,5)(2,3,4)$ | $\zeta_{3}$ | |
| $1$ | $6$ | $(1,4,5,3,6,2)$ | $-\zeta_{3}$ | |
| $1$ | $6$ | $(1,2,6,3,5,4)$ | $\zeta_{3} + 1$ |