Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(2668\)\(\medspace = 2^{2} \cdot 23 \cdot 29 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.0.163719152.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Projective image: | $S_3$ |
| Projective field: | Galois closure of 3.1.2668.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 17 + 21\cdot 31 + 13\cdot 31^{2} + 14\cdot 31^{3} + 29\cdot 31^{4} + 31^{5} + 19\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 4 a + \left(22 a + 29\right)\cdot 31 + \left(4 a + 27\right)\cdot 31^{2} + \left(15 a + 29\right)\cdot 31^{3} + \left(30 a + 10\right)\cdot 31^{4} + \left(30 a + 26\right)\cdot 31^{5} + \left(24 a + 8\right)\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 27 a + 8 + \left(8 a + 7\right)\cdot 31 + \left(26 a + 15\right)\cdot 31^{2} + \left(15 a + 24\right)\cdot 31^{3} + 25\cdot 31^{4} + 26\cdot 31^{5} + \left(6 a + 27\right)\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 4 a + 24 + \left(22 a + 23\right)\cdot 31 + \left(4 a + 15\right)\cdot 31^{2} + \left(15 a + 6\right)\cdot 31^{3} + \left(30 a + 5\right)\cdot 31^{4} + \left(30 a + 4\right)\cdot 31^{5} + \left(24 a + 3\right)\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 15 + 9\cdot 31 + 17\cdot 31^{2} + 16\cdot 31^{3} + 31^{4} + 29\cdot 31^{5} + 11\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 27 a + 1 + \left(8 a + 2\right)\cdot 31 + \left(26 a + 3\right)\cdot 31^{2} + \left(15 a + 1\right)\cdot 31^{3} + 20\cdot 31^{4} + 4\cdot 31^{5} + \left(6 a + 22\right)\cdot 31^{6} +O(31^{7})\)
|
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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,6)(3,4)$ | $-2$ |
| $3$ | $2$ | $(1,2)(5,6)$ | $0$ |
| $3$ | $2$ | $(1,5)(2,4)(3,6)$ | $0$ |
| $2$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $2$ | $6$ | $(1,4,2,5,3,6)$ | $1$ |