Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(3800\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 19 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.23104000.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Determinant: | 1.152.2t1.b.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.152.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 8x^{3} - 25x^{2} + 20x - 4 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 a + 7 + \left(25 a + 24\right)\cdot 31 + \left(25 a + 10\right)\cdot 31^{2} + \left(15 a + 24\right)\cdot 31^{3} + \left(3 a + 16\right)\cdot 31^{4} + \left(3 a + 23\right)\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 19 + 2\cdot 31 + 16\cdot 31^{2} + 27\cdot 31^{4} + 16\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 25 a + 19 + \left(5 a + 6\right)\cdot 31 + \left(5 a + 6\right)\cdot 31^{2} + \left(15 a + 30\right)\cdot 31^{3} + \left(27 a + 7\right)\cdot 31^{4} + \left(27 a + 26\right)\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 2 a + 19 + \left(27 a + 26\right)\cdot 31 + \left(29 a + 14\right)\cdot 31^{2} + \left(3 a + 8\right)\cdot 31^{3} + \left(22 a + 20\right)\cdot 31^{4} + \left(24 a + 30\right)\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 6 + 16\cdot 31 + 28\cdot 31^{2} + 11\cdot 31^{3} + 22\cdot 31^{4} + 30\cdot 31^{5} +O(31^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 29 a + 23 + \left(3 a + 16\right)\cdot 31 + \left(a + 16\right)\cdot 31^{2} + \left(27 a + 17\right)\cdot 31^{3} + \left(8 a + 29\right)\cdot 31^{4} + \left(6 a + 26\right)\cdot 31^{5} +O(31^{6})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-2$ |
| $3$ | $2$ | $(1,2)(5,6)$ | $0$ |
| $3$ | $2$ | $(1,6)(2,4)(3,5)$ | $0$ |
| $2$ | $3$ | $(1,2,3)(4,6,5)$ | $-1$ |
| $2$ | $6$ | $(1,4,2,6,3,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.