Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(66112\)\(\medspace = 2^{6} \cdot 1033 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.8818423496.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_3^2:D_4$ |
| Parity: | even |
| Determinant: | 1.1033.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.2.8818423496.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} - 9x^{4} + x^{3} + 34x^{2} + 45x - 238 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$:
\( x^{2} + 38x + 6 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 29 a + 26 + \left(5 a + 4\right)\cdot 41 + \left(21 a + 16\right)\cdot 41^{2} + \left(26 a + 16\right)\cdot 41^{3} + \left(30 a + 18\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 24 + 20\cdot 41 + 9\cdot 41^{2} + 15\cdot 41^{3} + 26\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 26 + 2\cdot 41 + 33\cdot 41^{2} + 31\cdot 41^{3} + 20\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 12 a + 31 + \left(35 a + 33\right)\cdot 41 + \left(19 a + 32\right)\cdot 41^{2} + \left(14 a + 33\right)\cdot 41^{3} + \left(10 a + 1\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 39 a + 12 + \left(30 a + 24\right)\cdot 41 + \left(11 a + 13\right)\cdot 41^{2} + \left(10 a + 3\right)\cdot 41^{3} + \left(24 a + 17\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 2 a + 6 + \left(10 a + 37\right)\cdot 41 + \left(29 a + 17\right)\cdot 41^{2} + \left(30 a + 22\right)\cdot 41^{3} + \left(16 a + 38\right)\cdot 41^{4} +O(41^{5})\)
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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 |
| $1$ | $1$ | $()$ | $4$ | |
| $6$ | $2$ | $(1,2)(3,5)(4,6)$ | $2$ | |
| $6$ | $2$ | $(1,3)$ | $0$ | |
| $9$ | $2$ | $(1,3)(2,5)$ | $0$ | ✓ |
| $4$ | $3$ | $(1,3,4)(2,5,6)$ | $1$ | |
| $4$ | $3$ | $(2,5,6)$ | $-2$ | |
| $18$ | $4$ | $(1,5,3,2)(4,6)$ | $0$ | |
| $12$ | $6$ | $(1,2,3,5,4,6)$ | $-1$ | |
| $12$ | $6$ | $(1,3)(2,5,6)$ | $0$ |