Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(1216\)\(\medspace = 2^{6} \cdot 19 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.0.739328.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Projective image: | $S_3$ |
| Projective field: | Galois closure of 3.1.76.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$:
\( x^{2} + 38x + 6 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 4 + 10\cdot 41 + 2\cdot 41^{2} + 28\cdot 41^{3} +O(41^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 22 a + 32 + \left(38 a + 8\right)\cdot 41 + \left(9 a + 2\right)\cdot 41^{2} + \left(7 a + 5\right)\cdot 41^{3} + \left(19 a + 4\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 19 a + 16 + \left(2 a + 20\right)\cdot 41 + \left(31 a + 34\right)\cdot 41^{2} + \left(33 a + 16\right)\cdot 41^{3} + \left(21 a + 13\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 37 + 30\cdot 41 + 38\cdot 41^{2} + 12\cdot 41^{3} + 40\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 19 a + 9 + \left(2 a + 32\right)\cdot 41 + \left(31 a + 38\right)\cdot 41^{2} + \left(33 a + 35\right)\cdot 41^{3} + \left(21 a + 36\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 22 a + 25 + \left(38 a + 20\right)\cdot 41 + \left(9 a + 6\right)\cdot 41^{2} + \left(7 a + 24\right)\cdot 41^{3} + \left(19 a + 27\right)\cdot 41^{4} +O(41^{5})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-2$ |
| $3$ | $2$ | $(2,3)(5,6)$ | $0$ |
| $3$ | $2$ | $(1,4)(2,6)(3,5)$ | $0$ |
| $2$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $2$ | $6$ | $(1,5,3,4,2,6)$ | $1$ |