Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(4739311\)\(\medspace = 17^{2} \cdot 23^{2} \cdot 31 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.1780800847561.4 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Determinant: | 1.31.2t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.31.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 197x^{4} + 684x^{3} + 8724x^{2} + 86240x - 28908921 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 28 + 6\cdot 29 + 5\cdot 29^{2} + 15\cdot 29^{3} + 5\cdot 29^{4} + 10\cdot 29^{5} + 6\cdot 29^{6} + 9\cdot 29^{7} + 25\cdot 29^{8} + 5\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 21 + 15\cdot 29 + 20\cdot 29^{2} + 5\cdot 29^{3} + 16\cdot 29^{4} + 29^{5} + 12\cdot 29^{6} + 25\cdot 29^{7} + 11\cdot 29^{8} + 24\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 27 a + 6 + \left(20 a + 1\right)\cdot 29 + \left(16 a + 24\right)\cdot 29^{2} + \left(14 a + 7\right)\cdot 29^{3} + \left(19 a + 28\right)\cdot 29^{4} + \left(19 a + 27\right)\cdot 29^{5} + \left(2 a + 28\right)\cdot 29^{6} + \left(13 a + 21\right)\cdot 29^{7} + \left(14 a + 15\right)\cdot 29^{8} + \left(21 a + 8\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 27 a + 24 + \left(13 a + 28\right)\cdot 29 + 21 a\cdot 29^{2} + \left(27 a + 11\right)\cdot 29^{3} + \left(8 a + 12\right)\cdot 29^{4} + \left(18 a + 1\right)\cdot 29^{5} + \left(8 a + 25\right)\cdot 29^{6} + \left(17 a + 20\right)\cdot 29^{7} + \left(16 a + 4\right)\cdot 29^{8} + \left(11 a + 25\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 2 a + 14 + \left(15 a + 13\right)\cdot 29 + \left(7 a + 7\right)\cdot 29^{2} + \left(a + 12\right)\cdot 29^{3} + 20 a\cdot 29^{4} + \left(10 a + 26\right)\cdot 29^{5} + \left(20 a + 20\right)\cdot 29^{6} + \left(11 a + 11\right)\cdot 29^{7} + \left(12 a + 12\right)\cdot 29^{8} + \left(17 a + 8\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 2 a + 25 + \left(8 a + 20\right)\cdot 29 + \left(12 a + 28\right)\cdot 29^{2} + \left(14 a + 5\right)\cdot 29^{3} + \left(9 a + 24\right)\cdot 29^{4} + \left(9 a + 19\right)\cdot 29^{5} + \left(26 a + 22\right)\cdot 29^{6} + \left(15 a + 26\right)\cdot 29^{7} + \left(14 a + 16\right)\cdot 29^{8} + \left(7 a + 14\right)\cdot 29^{9} +O(29^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
| $3$ | $2$ | $(1,3)(2,4)$ | $0$ |
| $3$ | $2$ | $(1,4)(2,3)(5,6)$ | $0$ |
| $2$ | $3$ | $(1,6,3)(2,5,4)$ | $-1$ |
| $2$ | $6$ | $(1,5,3,2,6,4)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.