Basic invariants
Dimension: | $9$ |
Group: | $S_6$ |
Conductor: | \(35647020474031\)\(\medspace = 32911^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.0.32911.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_{6}$ |
Parity: | odd |
Determinant: | 1.32911.2t1.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 |
$1$ | $1$ | $()$ | $9$ |
$15$ | $2$ | $(1,2)(3,4)(5,6)$ | $3$ |
$15$ | $2$ | $(1,2)$ | $3$ |
$45$ | $2$ | $(1,2)(3,4)$ | $1$ |
$40$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
$40$ | $3$ | $(1,2,3)$ | $0$ |
$90$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ |
$90$ | $4$ | $(1,2,3,4)$ | $-1$ |
$144$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
$120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
$120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.