Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(14225\)\(\medspace = 5^{2} \cdot 569 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.71125.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_3^2:D_4$ |
| Parity: | even |
| Determinant: | 1.569.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.2.71125.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} - x^{4} + 3x^{3} - x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$:
\( x^{2} + 60x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 37 a + 9 + \left(56 a + 19\right)\cdot 61 + \left(58 a + 10\right)\cdot 61^{2} + \left(53 a + 23\right)\cdot 61^{3} + \left(57 a + 48\right)\cdot 61^{4} +O(61^{5})\)
| $r_{ 2 }$ |
$=$ |
\( 27 + 54\cdot 61 + 32\cdot 61^{2} + 22\cdot 61^{3} + 55\cdot 61^{4} +O(61^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 11 a + 12 + \left(21 a + 59\right)\cdot 61 + \left(19 a + 14\right)\cdot 61^{2} + \left(58 a + 30\right)\cdot 61^{3} + \left(25 a + 49\right)\cdot 61^{4} +O(61^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 24 a + 46 + \left(4 a + 38\right)\cdot 61 + \left(2 a + 12\right)\cdot 61^{2} + \left(7 a + 18\right)\cdot 61^{3} + \left(3 a + 52\right)\cdot 61^{4} +O(61^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 7 + 3\cdot 61 + 38\cdot 61^{2} + 19\cdot 61^{3} + 21\cdot 61^{4} +O(61^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 50 a + 23 + \left(39 a + 8\right)\cdot 61 + \left(41 a + 13\right)\cdot 61^{2} + \left(2 a + 8\right)\cdot 61^{3} + \left(35 a + 17\right)\cdot 61^{4} +O(61^{5})\)
| |
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 |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $6$ | $2$ | $(3,6)$ | $2$ |
| $9$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $4$ | $3$ | $(1,4,5)$ | $1$ |
| $4$ | $3$ | $(1,4,5)(2,3,6)$ | $-2$ |
| $18$ | $4$ | $(1,2)(3,5,6,4)$ | $0$ |
| $12$ | $6$ | $(1,3,4,6,5,2)$ | $0$ |
| $12$ | $6$ | $(1,4,5)(3,6)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.