Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(756\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 7 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.1714608.3 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Determinant: | 1.84.2t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.756.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{5} + 3x^{4} - 6x^{2} + 12 \)
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The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$:
\( x^{2} + 12x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 12 + 4\cdot 13 + 2\cdot 13^{3} + 11\cdot 13^{4} + 12\cdot 13^{5} +O(13^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 9 + 8\cdot 13 + 10\cdot 13^{2} + 6\cdot 13^{3} + 2\cdot 13^{4} + 13^{5} +O(13^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 9 a + 8 + \left(12 a + 9\right)\cdot 13 + 8\cdot 13^{2} + \left(4 a + 5\right)\cdot 13^{3} + \left(5 a + 9\right)\cdot 13^{4} + \left(10 a + 2\right)\cdot 13^{5} +O(13^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 4 a + 4 + \left(12 a + 10\right)\cdot 13^{2} + \left(8 a + 8\right)\cdot 13^{3} + \left(7 a + 10\right)\cdot 13^{4} + \left(2 a + 7\right)\cdot 13^{5} +O(13^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 10 a + 6 + \left(6 a + 9\right)\cdot 13 + \left(2 a + 6\right)\cdot 13^{2} + \left(2 a + 1\right)\cdot 13^{3} + \left(2 a + 9\right)\cdot 13^{4} + \left(6 a + 11\right)\cdot 13^{5} +O(13^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 3 a + 3 + \left(6 a + 6\right)\cdot 13 + \left(10 a + 2\right)\cdot 13^{2} + \left(10 a + 1\right)\cdot 13^{3} + \left(10 a + 9\right)\cdot 13^{4} + \left(6 a + 2\right)\cdot 13^{5} +O(13^{6})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,5)(4,6)$ | $-2$ |
| $3$ | $2$ | $(1,3)(2,5)$ | $0$ |
| $3$ | $2$ | $(1,5)(2,3)(4,6)$ | $0$ |
| $2$ | $3$ | $(1,4,3)(2,6,5)$ | $-1$ |
| $2$ | $6$ | $(1,6,3,2,4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.