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 6.2.33362176000.17 |
| 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
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} - 41x^{4} + 104x^{3} + 264x^{2} + 928x + 96 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 26 + 18\cdot 31 + 3\cdot 31^{2} + 21\cdot 31^{3} + 3\cdot 31^{4} + 31^{5} + 5\cdot 31^{6} + 18\cdot 31^{7} +O(31^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 5 a + 5 + 25\cdot 31 + \left(5 a + 9\right)\cdot 31^{2} + \left(8 a + 11\right)\cdot 31^{3} + 12\cdot 31^{4} + \left(24 a + 24\right)\cdot 31^{5} + \left(2 a + 6\right)\cdot 31^{6} + \left(20 a + 4\right)\cdot 31^{7} +O(31^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 26 a + 15 + \left(30 a + 20\right)\cdot 31 + \left(25 a + 19\right)\cdot 31^{2} + \left(22 a + 22\right)\cdot 31^{3} + \left(30 a + 4\right)\cdot 31^{4} + \left(6 a + 10\right)\cdot 31^{5} + \left(28 a + 19\right)\cdot 31^{6} + \left(10 a + 10\right)\cdot 31^{7} +O(31^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 5 a + 8 + 20\cdot 31 + \left(5 a + 5\right)\cdot 31^{2} + \left(8 a + 25\right)\cdot 31^{3} + 31^{4} + \left(24 a + 15\right)\cdot 31^{5} + \left(2 a + 6\right)\cdot 31^{6} + 20 a\cdot 31^{7} +O(31^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 23 + 23\cdot 31 + 7\cdot 31^{2} + 7\cdot 31^{3} + 14\cdot 31^{4} + 10\cdot 31^{5} + 5\cdot 31^{6} + 22\cdot 31^{7} +O(31^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( 26 a + 18 + \left(30 a + 15\right)\cdot 31 + \left(25 a + 15\right)\cdot 31^{2} + \left(22 a + 5\right)\cdot 31^{3} + \left(30 a + 25\right)\cdot 31^{4} + 6 a\cdot 31^{5} + \left(28 a + 19\right)\cdot 31^{6} + \left(10 a + 6\right)\cdot 31^{7} +O(31^{8})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,4)(3,6)$ | $-2$ |
| $3$ | $2$ | $(1,2)(3,6)(4,5)$ | $0$ |
| $3$ | $2$ | $(1,6)(3,5)$ | $0$ |
| $2$ | $3$ | $(1,4,6)(2,3,5)$ | $-1$ |
| $2$ | $6$ | $(1,3,4,5,6,2)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.