Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(20025\)\(\medspace = 3^{2} \cdot 5^{2} \cdot 89 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.100125.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_3^2:D_4$ |
| Parity: | even |
| Determinant: | 1.89.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.2.100125.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 4x^{4} - 4x^{3} + 2x^{2} + x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$:
\( x^{2} + 69x + 7 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 7 + 58\cdot 71 + 14\cdot 71^{2} + 19\cdot 71^{3} + 12\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 12 a + 52 + \left(48 a + 4\right)\cdot 71 + \left(32 a + 13\right)\cdot 71^{2} + \left(48 a + 17\right)\cdot 71^{3} + \left(60 a + 68\right)\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 27 a + 41 + \left(22 a + 68\right)\cdot 71 + \left(54 a + 55\right)\cdot 71^{2} + \left(15 a + 1\right)\cdot 71^{3} + \left(10 a + 27\right)\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 44 a + 24 + \left(48 a + 15\right)\cdot 71 + 16 a\cdot 71^{2} + \left(55 a + 50\right)\cdot 71^{3} + \left(60 a + 31\right)\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 15 + 48\cdot 71 + 27\cdot 71^{2} + 43\cdot 71^{3} + 3\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 59 a + 5 + \left(22 a + 18\right)\cdot 71 + \left(38 a + 30\right)\cdot 71^{2} + \left(22 a + 10\right)\cdot 71^{3} + \left(10 a + 70\right)\cdot 71^{4} +O(71^{5})\)
|
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 value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,5)(4,6)$ | $0$ |
| $6$ | $2$ | $(3,4)$ | $2$ |
| $9$ | $2$ | $(3,4)(5,6)$ | $0$ |
| $4$ | $3$ | $(1,3,4)$ | $1$ |
| $4$ | $3$ | $(1,3,4)(2,5,6)$ | $-2$ |
| $18$ | $4$ | $(1,2)(3,6,4,5)$ | $0$ |
| $12$ | $6$ | $(1,5,3,6,4,2)$ | $0$ |
| $12$ | $6$ | $(2,5,6)(3,4)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.