Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(37\) |
| Artin field: | Galois closure of 6.6.69343957.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{37}(27,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 15x^{4} + 28x^{3} + 15x^{2} - 38x - 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$:
\( x^{2} + 21x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 22 a + 17 + \left(16 a + 5\right)\cdot 23 + \left(21 a + 19\right)\cdot 23^{2} + \left(10 a + 4\right)\cdot 23^{3} + \left(14 a + 12\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 12 a + 18 + \left(2 a + 13\right)\cdot 23 + \left(13 a + 8\right)\cdot 23^{2} + 19 a\cdot 23^{3} + \left(15 a + 3\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 11 a + 19 + \left(20 a + 6\right)\cdot 23 + \left(9 a + 9\right)\cdot 23^{2} + \left(3 a + 3\right)\cdot 23^{3} + \left(7 a + 15\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 15 a + 20 + \left(18 a + 12\right)\cdot 23 + \left(18 a + 6\right)\cdot 23^{2} + \left(12 a + 1\right)\cdot 23^{3} + \left(12 a + 21\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 8 a + 4 + \left(4 a + 12\right)\cdot 23 + \left(4 a + 2\right)\cdot 23^{2} + \left(10 a + 8\right)\cdot 23^{3} + \left(10 a + 10\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( a + 15 + \left(6 a + 17\right)\cdot 23 + \left(a + 22\right)\cdot 23^{2} + \left(12 a + 4\right)\cdot 23^{3} + \left(8 a + 7\right)\cdot 23^{4} +O(23^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ | ✓ |
| $1$ | $2$ | $(1,6)(2,3)(4,5)$ | $-1$ | |
| $1$ | $3$ | $(1,5,3)(2,6,4)$ | $\zeta_{3}$ | |
| $1$ | $3$ | $(1,3,5)(2,4,6)$ | $-\zeta_{3} - 1$ | |
| $1$ | $6$ | $(1,2,5,6,3,4)$ | $\zeta_{3} + 1$ | |
| $1$ | $6$ | $(1,4,3,6,5,2)$ | $-\zeta_{3}$ |