Basic invariants
| Dimension: | $5$ |
| Group: | $S_6$ |
| Conductor: | \(47669\)\(\medspace = 73 \cdot 653 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.47669.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_6$ |
| Parity: | even |
| Determinant: | 1.47669.2t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.2.47669.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + x^{4} - x^{2} - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 397 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 397 }$:
\( x^{2} + 392x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 282 + 305\cdot 397 + 79\cdot 397^{2} + 80\cdot 397^{3} + 121\cdot 397^{4} +O(397^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 301 + 164\cdot 397 + 135\cdot 397^{2} + 362\cdot 397^{3} + 56\cdot 397^{4} +O(397^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 131 a + 373 + \left(208 a + 189\right)\cdot 397 + \left(321 a + 333\right)\cdot 397^{2} + 3 a\cdot 397^{3} + \left(49 a + 79\right)\cdot 397^{4} +O(397^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 364 a + 282 + \left(96 a + 51\right)\cdot 397 + \left(71 a + 119\right)\cdot 397^{2} + \left(311 a + 179\right)\cdot 397^{3} + \left(127 a + 341\right)\cdot 397^{4} +O(397^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 266 a + 234 + \left(188 a + 306\right)\cdot 397 + \left(75 a + 144\right)\cdot 397^{2} + \left(393 a + 95\right)\cdot 397^{3} + \left(347 a + 320\right)\cdot 397^{4} +O(397^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 33 a + 117 + \left(300 a + 172\right)\cdot 397 + \left(325 a + 378\right)\cdot 397^{2} + \left(85 a + 75\right)\cdot 397^{3} + \left(269 a + 272\right)\cdot 397^{4} +O(397^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
| $15$ | $2$ | $(1,2)$ | $3$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $2$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.