Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(755\)\(\medspace = 5 \cdot 151 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.2.2850125.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Projective image: | $S_3$ |
| Projective field: | Galois closure of 3.1.755.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$:
\( x^{2} + 38x + 6 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 29 + 24\cdot 41 + 41^{2} + 20\cdot 41^{3} + 6\cdot 41^{4} + 15\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 36 a + 19 + \left(29 a + 15\right)\cdot 41 + \left(24 a + 30\right)\cdot 41^{2} + \left(14 a + 15\right)\cdot 41^{3} + \left(3 a + 32\right)\cdot 41^{4} + \left(5 a + 10\right)\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 36 a + 38 + \left(29 a + 12\right)\cdot 41 + \left(24 a + 7\right)\cdot 41^{2} + \left(14 a + 6\right)\cdot 41^{3} + \left(3 a + 13\right)\cdot 41^{4} + \left(5 a + 18\right)\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 5 a + 4 + \left(11 a + 28\right)\cdot 41 + \left(16 a + 33\right)\cdot 41^{2} + \left(26 a + 34\right)\cdot 41^{3} + \left(37 a + 27\right)\cdot 41^{4} + \left(35 a + 22\right)\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 13 + 16\cdot 41 + 39\cdot 41^{2} + 20\cdot 41^{3} + 34\cdot 41^{4} + 25\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 5 a + 23 + \left(11 a + 25\right)\cdot 41 + \left(16 a + 10\right)\cdot 41^{2} + \left(26 a + 25\right)\cdot 41^{3} + \left(37 a + 8\right)\cdot 41^{4} + \left(35 a + 30\right)\cdot 41^{5} +O(41^{6})\)
|
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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,6)(3,4)$ | $-2$ |
| $3$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $3$ | $2$ | $(1,3)(4,5)$ | $0$ |
| $2$ | $3$ | $(1,6,3)(2,4,5)$ | $-1$ |
| $2$ | $6$ | $(1,4,6,5,3,2)$ | $1$ |