Basic invariants
| Dimension: | $4$ |
| Group: | $C_3^2:D_4$ |
| Conductor: | \(10225\)\(\medspace = 5^{2} \cdot 409 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.51125.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_3^2:D_4$ |
| Parity: | even |
| Determinant: | 1.409.2t1.a.a |
| Projective image: | $\SOPlus(4,2)$ |
| Projective stem field: | Galois closure of 6.2.51125.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 2x^{4} - 2x^{2} + 3x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 109 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 109 }$:
\( x^{2} + 108x + 6 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 35 + 32\cdot 109 + 72\cdot 109^{2} + 55\cdot 109^{3} + 82\cdot 109^{4} +O(109^{5})\)
| $r_{ 2 }$ |
$=$ |
\( 98 a + 49 + \left(47 a + 68\right)\cdot 109 + \left(80 a + 78\right)\cdot 109^{2} + \left(48 a + 91\right)\cdot 109^{3} + \left(80 a + 73\right)\cdot 109^{4} +O(109^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 71 a + 2 + \left(80 a + 88\right)\cdot 109 + \left(33 a + 41\right)\cdot 109^{2} + \left(74 a + 6\right)\cdot 109^{3} + \left(44 a + 28\right)\cdot 109^{4} +O(109^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 23 + 22\cdot 109 + 28\cdot 109^{2} + 66\cdot 109^{3} + 38\cdot 109^{4} +O(109^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 11 a + 38 + \left(61 a + 18\right)\cdot 109 + \left(28 a + 2\right)\cdot 109^{2} + \left(60 a + 60\right)\cdot 109^{3} + \left(28 a + 105\right)\cdot 109^{4} +O(109^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 38 a + 73 + \left(28 a + 97\right)\cdot 109 + \left(75 a + 103\right)\cdot 109^{2} + \left(34 a + 46\right)\cdot 109^{3} + \left(64 a + 107\right)\cdot 109^{4} +O(109^{5})\)
| |
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 |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $6$ | $2$ | $(3,6)$ | $2$ |
| $9$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $4$ | $3$ | $(1,3,6)$ | $1$ |
| $4$ | $3$ | $(1,3,6)(2,4,5)$ | $-2$ |
| $18$ | $4$ | $(1,2)(3,5,6,4)$ | $0$ |
| $12$ | $6$ | $(1,4,3,5,6,2)$ | $0$ |
| $12$ | $6$ | $(2,4,5)(3,6)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.