Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(153\)\(\medspace = 3^{2} \cdot 17 \) |
| Artin field: | Galois closure of 6.0.96702579.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{153}(50,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + 24x^{4} - 13x^{3} + 144x^{2} - 156x + 361 \)
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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:
| $r_{ 1 }$ | $=$ |
\( a + 5 + \left(11 a + 26\right)\cdot 37 + \left(34 a + 28\right)\cdot 37^{2} + \left(11 a + 8\right)\cdot 37^{3} + \left(14 a + 35\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 15 a + 23 + \left(a + 31\right)\cdot 37 + \left(4 a + 8\right)\cdot 37^{2} + \left(20 a + 29\right)\cdot 37^{3} + 2 a\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 14 a + 23 + \left(27 a + 25\right)\cdot 37 + \left(6 a + 21\right)\cdot 37^{2} + \left(8 a + 33\right)\cdot 37^{3} + \left(25 a + 10\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 36 a + 9 + \left(25 a + 32\right)\cdot 37 + \left(2 a + 6\right)\cdot 37^{2} + \left(25 a + 22\right)\cdot 37^{3} + \left(22 a + 6\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 22 a + 9 + \left(35 a + 22\right)\cdot 37 + \left(32 a + 23\right)\cdot 37^{2} + \left(16 a + 31\right)\cdot 37^{3} + \left(34 a + 27\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 23 a + 5 + \left(9 a + 10\right)\cdot 37 + \left(30 a + 21\right)\cdot 37^{2} + \left(28 a + 22\right)\cdot 37^{3} + \left(11 a + 29\right)\cdot 37^{4} +O(37^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ |
| $1$ | $3$ | $(1,5,3)(2,6,4)$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,3,5)(2,4,6)$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,6,5,4,3,2)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,2,3,4,5,6)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.