Basic invariants
Dimension: | $10$ |
Group: | $S_6$ |
Conductor: | \(5278839342209655121\)\(\medspace = 47933^{4} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.2.47933.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 30T164 |
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 |
$1$ | $1$ | $()$ | $10$ |
$15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
$15$ | $2$ | $(1,2)$ | $2$ |
$45$ | $2$ | $(1,2)(3,4)$ | $-2$ |
$40$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
$40$ | $3$ | $(1,2,3)$ | $1$ |
$90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
$90$ | $4$ | $(1,2,3,4)$ | $0$ |
$144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
$120$ | $6$ | $(1,2,3,4,5,6)$ | $1$ |
$120$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.