Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(104\)\(\medspace = 2^{3} \cdot 13 \) |
| Artin field: | Galois closure of 6.0.14623232.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{104}(35,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} - x^{4} - 2x^{3} + 34x^{2} + 28x + 73 \)
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The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( a + 3 + \left(23 a + 30\right)\cdot 31 + \left(12 a + 29\right)\cdot 31^{2} + \left(23 a + 23\right)\cdot 31^{3} + \left(6 a + 25\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 30 a + 23 + \left(7 a + 26\right)\cdot 31 + \left(18 a + 25\right)\cdot 31^{2} + \left(7 a + 7\right)\cdot 31^{3} + \left(24 a + 8\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 30 a + 7 + \left(7 a + 12\right)\cdot 31 + \left(18 a + 7\right)\cdot 31^{2} + \left(7 a + 16\right)\cdot 31^{3} + \left(24 a + 7\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( a + 5 + \left(23 a + 29\right)\cdot 31 + \left(12 a + 4\right)\cdot 31^{2} + \left(23 a + 13\right)\cdot 31^{3} + \left(6 a + 17\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( a + 21 + \left(23 a + 12\right)\cdot 31 + \left(12 a + 23\right)\cdot 31^{2} + \left(23 a + 4\right)\cdot 31^{3} + \left(6 a + 18\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 30 a + 5 + \left(7 a + 13\right)\cdot 31 + \left(18 a + 1\right)\cdot 31^{2} + \left(7 a + 27\right)\cdot 31^{3} + \left(24 a + 15\right)\cdot 31^{4} +O(31^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ | |
| $1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-1$ | ✓ |
| $1$ | $3$ | $(1,4,5)(2,6,3)$ | $-\zeta_{3} - 1$ | |
| $1$ | $3$ | $(1,5,4)(2,3,6)$ | $\zeta_{3}$ | |
| $1$ | $6$ | $(1,2,4,6,5,3)$ | $-\zeta_{3}$ | |
| $1$ | $6$ | $(1,3,5,6,4,2)$ | $\zeta_{3} + 1$ |