Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(5585854293\)\(\medspace = 3^{2} \cdot 853^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.4.1861951431.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T34 |
| Parity: | even |
| Determinant: | 1.853.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.4.1861951431.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + x^{4} - 29x^{3} + 29x^{2} - 3 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$:
\( x^{2} + 70x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 40 + 19\cdot 73 + 66\cdot 73^{2} + 3\cdot 73^{3} + 22\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 8 a + 5 + \left(40 a + 7\right)\cdot 73 + \left(60 a + 42\right)\cdot 73^{2} + \left(61 a + 8\right)\cdot 73^{3} + \left(70 a + 23\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 16 + 73 + 65\cdot 73^{2} + 15\cdot 73^{3} + 52\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 63 a + 44 + \left(72 a + 67\right)\cdot 73 + \left(71 a + 41\right)\cdot 73^{2} + \left(62 a + 6\right)\cdot 73^{3} + \left(41 a + 52\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 65 a + 29 + \left(32 a + 46\right)\cdot 73 + \left(12 a + 37\right)\cdot 73^{2} + \left(11 a + 60\right)\cdot 73^{3} + \left(2 a + 27\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 10 a + 14 + 4\cdot 73 + \left(a + 39\right)\cdot 73^{2} + \left(10 a + 50\right)\cdot 73^{3} + \left(31 a + 41\right)\cdot 73^{4} +O(73^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,3)(2,4)(5,6)$ | $-2$ |
| $6$ | $2$ | $(4,6)$ | $0$ |
| $9$ | $2$ | $(2,5)(4,6)$ | $0$ |
| $4$ | $3$ | $(3,4,6)$ | $-2$ |
| $4$ | $3$ | $(1,2,5)(3,4,6)$ | $1$ |
| $18$ | $4$ | $(1,3)(2,4,5,6)$ | $0$ |
| $12$ | $6$ | $(1,3,2,4,5,6)$ | $1$ |
| $12$ | $6$ | $(1,2,5)(4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.