Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(1960\)\(\medspace = 2^{3} \cdot 5 \cdot 7^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.26891200.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Determinant: | 1.40.2t1.b.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.1960.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 11x^{4} - 8x^{3} + 30x^{2} + 24x + 8 \)
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The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 16 + 25\cdot 29 + 19\cdot 29^{2} + 29^{3} + 23\cdot 29^{4} + 22\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 15 a + 13 + \left(19 a + 18\right)\cdot 29 + \left(10 a + 16\right)\cdot 29^{2} + \left(6 a + 17\right)\cdot 29^{3} + \left(28 a + 22\right)\cdot 29^{4} + \left(22 a + 17\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 12 + 12\cdot 29 + 22\cdot 29^{2} + 8\cdot 29^{3} + 23\cdot 29^{4} + 25\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 14 a + 1 + \left(9 a + 14\right)\cdot 29 + \left(18 a + 21\right)\cdot 29^{2} + \left(22 a + 9\right)\cdot 29^{3} + 12\cdot 29^{4} + \left(6 a + 17\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 27 a + 14 + \left(17 a + 20\right)\cdot 29 + \left(15 a + 16\right)\cdot 29^{2} + \left(25 a + 26\right)\cdot 29^{3} + \left(27 a + 3\right)\cdot 29^{4} + \left(14 a + 7\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 2 a + 4 + \left(11 a + 25\right)\cdot 29 + \left(13 a + 18\right)\cdot 29^{2} + \left(3 a + 22\right)\cdot 29^{3} + \left(a + 1\right)\cdot 29^{4} + \left(14 a + 25\right)\cdot 29^{5} +O(29^{6})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,5)(4,6)$ | $-2$ |
| $3$ | $2$ | $(1,2)(3,5)$ | $0$ |
| $3$ | $2$ | $(1,5)(2,3)(4,6)$ | $0$ |
| $2$ | $3$ | $(1,4,2)(3,6,5)$ | $-1$ |
| $2$ | $6$ | $(1,6,2,3,4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.