Basic invariants
| Dimension: | $5$ |
| Group: | $S_6$ |
| Conductor: | \(19125783616\)\(\medspace = 2^{6} \cdot 59^{2} \cdot 293^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.138296.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T183 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.0.138296.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 2x^{4} + 2x^{2} + 3x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 439 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 439 }$:
\( x^{2} + 436x + 15 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 21 + 426\cdot 439 + 256\cdot 439^{2} + 43\cdot 439^{3} + 300\cdot 439^{4} +O(439^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 173 + 62\cdot 439 + 230\cdot 439^{2} + 192\cdot 439^{3} + 32\cdot 439^{4} +O(439^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 177 a + 18 + \left(411 a + 430\right)\cdot 439 + \left(209 a + 12\right)\cdot 439^{2} + \left(129 a + 265\right)\cdot 439^{3} + \left(274 a + 3\right)\cdot 439^{4} +O(439^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 262 a + 110 + \left(27 a + 170\right)\cdot 439 + \left(229 a + 231\right)\cdot 439^{2} + \left(309 a + 4\right)\cdot 439^{3} + \left(164 a + 258\right)\cdot 439^{4} +O(439^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 192 + 377\cdot 439 + 397\cdot 439^{2} + 410\cdot 439^{3} + 121\cdot 439^{4} +O(439^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 365 + 289\cdot 439 + 187\cdot 439^{2} + 400\cdot 439^{3} + 161\cdot 439^{4} +O(439^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $15$ | $2$ | $(1,2)$ | $1$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
| $40$ | $3$ | $(1,2,3)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.