Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(57760\)\(\medspace = 2^{5} \cdot 5 \cdot 19^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.16681088000.5 |
| 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 3.1.14440.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 11x^{4} + 224x^{2} + 1280 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 a + 9 + \left(21 a + 28\right)\cdot 29 + \left(13 a + 2\right)\cdot 29^{2} + 11\cdot 29^{3} + \left(9 a + 9\right)\cdot 29^{4} + \left(22 a + 20\right)\cdot 29^{5} + \left(16 a + 28\right)\cdot 29^{6} + \left(11 a + 7\right)\cdot 29^{7} +O(29^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 12 + 12\cdot 29 + 2\cdot 29^{2} + 5\cdot 29^{3} + 10\cdot 29^{4} + 10\cdot 29^{5} + 14\cdot 29^{6} + 13\cdot 29^{7} +O(29^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 3 a + 5 + \left(21 a + 14\right)\cdot 29 + \left(13 a + 7\right)\cdot 29^{2} + \left(9 a + 4\right)\cdot 29^{4} + \left(22 a + 22\right)\cdot 29^{5} + \left(16 a + 25\right)\cdot 29^{6} + \left(11 a + 8\right)\cdot 29^{7} +O(29^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 26 a + 20 + 7 a\cdot 29 + \left(15 a + 26\right)\cdot 29^{2} + \left(28 a + 17\right)\cdot 29^{3} + \left(19 a + 19\right)\cdot 29^{4} + \left(6 a + 8\right)\cdot 29^{5} + 12 a\cdot 29^{6} + \left(17 a + 21\right)\cdot 29^{7} +O(29^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 17 + 16\cdot 29 + 26\cdot 29^{2} + 23\cdot 29^{3} + 18\cdot 29^{4} + 18\cdot 29^{5} + 14\cdot 29^{6} + 15\cdot 29^{7} +O(29^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( 26 a + 24 + \left(7 a + 14\right)\cdot 29 + \left(15 a + 21\right)\cdot 29^{2} + \left(28 a + 28\right)\cdot 29^{3} + \left(19 a + 24\right)\cdot 29^{4} + \left(6 a + 6\right)\cdot 29^{5} + \left(12 a + 3\right)\cdot 29^{6} + \left(17 a + 20\right)\cdot 29^{7} +O(29^{8})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-2$ |
| $3$ | $2$ | $(1,2)(4,5)$ | $0$ |
| $3$ | $2$ | $(1,4)(2,3)(5,6)$ | $0$ |
| $2$ | $3$ | $(1,2,6)(3,4,5)$ | $-1$ |
| $2$ | $6$ | $(1,3,2,4,6,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.