Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(231\)\(\medspace = 3 \cdot 7 \cdot 11 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.0.160083.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Projective image: | $S_3$ |
| Projective field: | Galois closure of 3.1.231.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 16\cdot 31 + 4\cdot 31^{2} + 14\cdot 31^{3} + 18\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 17 a + 5 + \left(21 a + 26\right)\cdot 31 + \left(25 a + 8\right)\cdot 31^{2} + \left(5 a + 22\right)\cdot 31^{3} + \left(10 a + 2\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 14 a + 8 + \left(9 a + 21\right)\cdot 31 + \left(5 a + 7\right)\cdot 31^{2} + \left(25 a + 8\right)\cdot 31^{3} + \left(20 a + 17\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 8 a + 6 + \left(26 a + 16\right)\cdot 31 + \left(29 a + 27\right)\cdot 31^{2} + \left(11 a + 26\right)\cdot 31^{3} + \left(12 a + 30\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 23 a + 22 + \left(4 a + 29\right)\cdot 31 + \left(a + 29\right)\cdot 31^{2} + \left(19 a + 20\right)\cdot 31^{3} + \left(18 a + 12\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 18 + 14\cdot 31 + 14\cdot 31^{2} + 11\cdot 31^{4} +O(31^{5})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,4)(3,5)$ | $-2$ |
| $3$ | $2$ | $(1,2)(3,5)(4,6)$ | $0$ |
| $3$ | $2$ | $(1,5)(3,6)$ | $0$ |
| $2$ | $3$ | $(1,4,5)(2,3,6)$ | $-1$ |
| $2$ | $6$ | $(1,3,4,6,5,2)$ | $1$ |