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