Basic invariants
| Dimension: | $5$ |
| Group: | $S_6$ |
| Conductor: | \(2297572489\)\(\medspace = 47933^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.2.47933.1 |
| 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 6.2.47933.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 2x^{4} - x^{3} - 1 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 82 a + 7 + \left(41 a + 33\right)\cdot 109 + \left(70 a + 36\right)\cdot 109^{2} + \left(13 a + 104\right)\cdot 109^{3} + \left(76 a + 97\right)\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 80 a + 41 + \left(101 a + 78\right)\cdot 109 + \left(21 a + 102\right)\cdot 109^{2} + \left(89 a + 5\right)\cdot 109^{3} + \left(97 a + 35\right)\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 29 a + 12 + \left(7 a + 100\right)\cdot 109 + \left(87 a + 22\right)\cdot 109^{2} + \left(19 a + 73\right)\cdot 109^{3} + \left(11 a + 43\right)\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 27 a + 89 + \left(67 a + 101\right)\cdot 109 + \left(38 a + 64\right)\cdot 109^{2} + \left(95 a + 47\right)\cdot 109^{3} + \left(32 a + 51\right)\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 21 + 102\cdot 109 + 63\cdot 109^{2} + 60\cdot 109^{3} + 60\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 50 + 20\cdot 109 + 36\cdot 109^{2} + 35\cdot 109^{3} + 38\cdot 109^{4} +O(109^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $15$ | $2$ | $(1,2)$ | $1$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
| $40$ | $3$ | $(1,2,3)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.