Basic invariants
Dimension: | $10$ |
Group: | $S_6$ |
Conductor: | \(13947137604000000\)\(\medspace = 2^{8} \cdot 3^{20} \cdot 5^{6} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.2.4723920.2 |
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 6.2.4723920.2 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} + 3x^{4} - 4x^{3} + 9x^{2} - 6x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 149 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 149 }$: \( x^{2} + 145x + 2 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 102 a + 4 + \left(126 a + 81\right)\cdot 149 + \left(113 a + 123\right)\cdot 149^{2} + \left(88 a + 112\right)\cdot 149^{3} + \left(125 a + 8\right)\cdot 149^{4} +O(149^{5})\)
$r_{ 2 }$ |
$=$ |
\( 47 a + 114 + \left(22 a + 38\right)\cdot 149 + \left(35 a + 5\right)\cdot 149^{2} + \left(60 a + 56\right)\cdot 149^{3} + \left(23 a + 124\right)\cdot 149^{4} +O(149^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 87 a + 64 + \left(103 a + 96\right)\cdot 149 + \left(57 a + 98\right)\cdot 149^{2} + \left(62 a + 136\right)\cdot 149^{3} + \left(125 a + 85\right)\cdot 149^{4} +O(149^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 62 a + 114 + \left(45 a + 125\right)\cdot 149 + \left(91 a + 76\right)\cdot 149^{2} + \left(86 a + 30\right)\cdot 149^{3} + \left(23 a + 78\right)\cdot 149^{4} +O(149^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 104 + 57\cdot 149 + 6\cdot 149^{2} + 90\cdot 149^{3} + 85\cdot 149^{4} +O(149^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 47 + 47\cdot 149 + 136\cdot 149^{2} + 20\cdot 149^{3} + 64\cdot 149^{4} +O(149^{5})\)
| |
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.