Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(2024\)\(\medspace = 2^{3} \cdot 11 \cdot 23 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.94221248.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | even |
| Determinant: | 1.2024.2t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.3.2024.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{5} + 15x^{4} - 25x^{3} + 34x^{2} - 22x + 8 \)
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The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$:
\( x^{2} + 38x + 6 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 39 a + 5 + \left(4 a + 10\right)\cdot 41 + \left(3 a + 1\right)\cdot 41^{2} + \left(22 a + 2\right)\cdot 41^{3} + \left(16 a + 6\right)\cdot 41^{4} + \left(37 a + 16\right)\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 2 a + 40 + \left(36 a + 26\right)\cdot 41 + \left(37 a + 5\right)\cdot 41^{2} + \left(18 a + 24\right)\cdot 41^{3} + \left(24 a + 33\right)\cdot 41^{4} + \left(3 a + 29\right)\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 19 + 13\cdot 41 + 29\cdot 41^{2} + 13\cdot 41^{3} + 40\cdot 41^{4} + 16\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 2 a + 37 + \left(36 a + 30\right)\cdot 41 + \left(37 a + 39\right)\cdot 41^{2} + \left(18 a + 38\right)\cdot 41^{3} + \left(24 a + 34\right)\cdot 41^{4} + \left(3 a + 24\right)\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 39 a + 2 + \left(4 a + 14\right)\cdot 41 + \left(3 a + 35\right)\cdot 41^{2} + \left(22 a + 16\right)\cdot 41^{3} + \left(16 a + 7\right)\cdot 41^{4} + \left(37 a + 11\right)\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 23 + 27\cdot 41 + 11\cdot 41^{2} + 27\cdot 41^{3} + 24\cdot 41^{5} +O(41^{6})\)
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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 | Complex conjugation |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-2$ | ✓ |
| $3$ | $2$ | $(1,2)(4,5)$ | $0$ | |
| $3$ | $2$ | $(1,3)(2,5)(4,6)$ | $0$ | |
| $2$ | $3$ | $(1,6,2)(3,5,4)$ | $-1$ | |
| $2$ | $6$ | $(1,5,6,4,2,3)$ | $1$ |