Basic invariants
Dimension: | $16$ |
Group: | $S_6$ |
Conductor: | \(891\!\cdots\!000\)\(\medspace = 2^{36} \cdot 3^{12} \cdot 5^{12} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.2.25920000.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 36T1252 |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $S_6$ |
Projective stem field: | Galois closure of 6.2.25920000.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - 2x^{5} - 4x^{4} + 12x^{3} - 14x^{2} + 8x - 4 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 137 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 137 }$: \( x^{2} + 131x + 3 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 3 + 31\cdot 137 + 3\cdot 137^{2} + 16\cdot 137^{3} + 61\cdot 137^{4} +O(137^{5})\)
$r_{ 2 }$ |
$=$ |
\( 115 a + 71 + \left(136 a + 125\right)\cdot 137 + \left(121 a + 96\right)\cdot 137^{2} + \left(69 a + 39\right)\cdot 137^{3} + \left(26 a + 103\right)\cdot 137^{4} +O(137^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 11 a + 81 + \left(121 a + 18\right)\cdot 137 + \left(26 a + 58\right)\cdot 137^{2} + \left(78 a + 63\right)\cdot 137^{3} + \left(69 a + 121\right)\cdot 137^{4} +O(137^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 22 a + 76 + 9\cdot 137 + \left(15 a + 7\right)\cdot 137^{2} + \left(67 a + 63\right)\cdot 137^{3} + \left(110 a + 55\right)\cdot 137^{4} +O(137^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 126 a + 10 + \left(15 a + 49\right)\cdot 137 + \left(110 a + 98\right)\cdot 137^{2} + \left(58 a + 94\right)\cdot 137^{3} + \left(67 a + 49\right)\cdot 137^{4} +O(137^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 35 + 40\cdot 137 + 10\cdot 137^{2} + 134\cdot 137^{3} + 19\cdot 137^{4} +O(137^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $16$ |
$15$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
$15$ | $2$ | $(1,2)$ | $0$ |
$45$ | $2$ | $(1,2)(3,4)$ | $0$ |
$40$ | $3$ | $(1,2,3)(4,5,6)$ | $-2$ |
$40$ | $3$ | $(1,2,3)$ | $-2$ |
$90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
$90$ | $4$ | $(1,2,3,4)$ | $0$ |
$144$ | $5$ | $(1,2,3,4,5)$ | $1$ |
$120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
$120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.