Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(1900\)\(\medspace = 2^{2} \cdot 5^{2} \cdot 19 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.722000.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Determinant: | 1.19.2t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.76.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 6x^{4} - 8x^{3} + 4x^{2} + 4x - 4 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 27 a + 17 + \left(25 a + 20\right)\cdot 29 + \left(a + 2\right)\cdot 29^{2} + \left(3 a + 19\right)\cdot 29^{3} + \left(9 a + 28\right)\cdot 29^{4} + 19 a\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 21 a + 10 + \left(3 a + 4\right)\cdot 29 + \left(22 a + 18\right)\cdot 29^{2} + \left(12 a + 24\right)\cdot 29^{3} + \left(23 a + 6\right)\cdot 29^{4} + \left(27 a + 19\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 2 a + 7 + \left(3 a + 7\right)\cdot 29 + \left(27 a + 15\right)\cdot 29^{2} + \left(25 a + 3\right)\cdot 29^{3} + \left(19 a + 13\right)\cdot 29^{4} + \left(9 a + 1\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 8 a + 28 + \left(25 a + 1\right)\cdot 29 + \left(6 a + 9\right)\cdot 29^{2} + \left(16 a + 8\right)\cdot 29^{3} + \left(5 a + 24\right)\cdot 29^{4} + \left(a + 18\right)\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 5 + 29 + 11\cdot 29^{2} + 6\cdot 29^{3} + 16\cdot 29^{4} + 26\cdot 29^{5} +O(29^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 20 + 22\cdot 29 + 29^{2} + 25\cdot 29^{3} + 26\cdot 29^{4} + 19\cdot 29^{5} +O(29^{6})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
| $3$ | $2$ | $(1,3)(2,4)$ | $0$ |
| $3$ | $2$ | $(1,4)(2,3)(5,6)$ | $0$ |
| $2$ | $3$ | $(1,5,3)(2,6,4)$ | $-1$ |
| $2$ | $6$ | $(1,6,3,2,5,4)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.