Basic invariants
| Dimension: | $5$ |
| Group: | $S_6$ |
| Conductor: | \(1173179090820834241\)\(\medspace = 32911^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.32911.1 |
| 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 6.0.32911.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{3} - x^{2} + x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$:
\( x^{2} + 96x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 18 a + 42 + \left(13 a + 20\right)\cdot 97 + \left(27 a + 61\right)\cdot 97^{2} + \left(84 a + 2\right)\cdot 97^{3} + \left(a + 80\right)\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 33 a + 72 + \left(35 a + 60\right)\cdot 97 + \left(43 a + 52\right)\cdot 97^{2} + \left(61 a + 8\right)\cdot 97^{3} + \left(3 a + 63\right)\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 64 a + 8 + \left(61 a + 63\right)\cdot 97 + \left(53 a + 60\right)\cdot 97^{2} + \left(35 a + 26\right)\cdot 97^{3} + \left(93 a + 5\right)\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 62 a + 72 + \left(92 a + 1\right)\cdot 97 + \left(38 a + 96\right)\cdot 97^{2} + \left(96 a + 67\right)\cdot 97^{3} + \left(40 a + 51\right)\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 79 a + 60 + \left(83 a + 15\right)\cdot 97 + \left(69 a + 75\right)\cdot 97^{2} + \left(12 a + 59\right)\cdot 97^{3} + \left(95 a + 94\right)\cdot 97^{4} +O(97^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 35 a + 37 + \left(4 a + 32\right)\cdot 97 + \left(58 a + 42\right)\cdot 97^{2} + 28\cdot 97^{3} + \left(56 a + 93\right)\cdot 97^{4} +O(97^{5})\)
|
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.