Basic invariants
Dimension: | $10$ |
Group: | $S_6$ |
Conductor: | \(4347986536861696\)\(\medspace = 2^{12} \cdot 101^{6} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 6.0.81608.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 30T164 |
Parity: | even |
Projective image: | $S_6$ |
Projective field: | Galois closure of 6.0.81608.1 |
Galois action
Roots of defining polynomial
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:
$r_{ 1 }$ | $=$ |
\( 29 a + 41 + \left(10 a + 26\right)\cdot 109 + \left(96 a + 10\right)\cdot 109^{2} + \left(60 a + 6\right)\cdot 109^{3} + \left(81 a + 90\right)\cdot 109^{4} +O(109^{5})\)
$r_{ 2 }$ |
$=$ |
\( 34 a + 69 + \left(25 a + 10\right)\cdot 109 + \left(98 a + 48\right)\cdot 109^{2} + \left(21 a + 83\right)\cdot 109^{3} + \left(20 a + 27\right)\cdot 109^{4} +O(109^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 96 a + 29 + \left(92 a + 87\right)\cdot 109 + \left(63 a + 94\right)\cdot 109^{2} + \left(32 a + 90\right)\cdot 109^{3} + \left(84 a + 64\right)\cdot 109^{4} +O(109^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 75 a + 103 + \left(83 a + 1\right)\cdot 109 + \left(10 a + 12\right)\cdot 109^{2} + \left(87 a + 7\right)\cdot 109^{3} + \left(88 a + 26\right)\cdot 109^{4} +O(109^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 13 a + 16 + \left(16 a + 84\right)\cdot 109 + \left(45 a + 65\right)\cdot 109^{2} + \left(76 a + 59\right)\cdot 109^{3} + \left(24 a + 7\right)\cdot 109^{4} +O(109^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 80 a + 70 + \left(98 a + 7\right)\cdot 109 + \left(12 a + 96\right)\cdot 109^{2} + \left(48 a + 79\right)\cdot 109^{3} + \left(27 a + 1\right)\cdot 109^{4} +O(109^{5})\)
| |
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 values |
$c1$ | |||
$1$ | $1$ | $()$ | $10$ |
$15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
$15$ | $2$ | $(1,2)$ | $2$ |
$45$ | $2$ | $(1,2)(3,4)$ | $-2$ |
$40$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
$40$ | $3$ | $(1,2,3)$ | $1$ |
$90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
$90$ | $4$ | $(1,2,3,4)$ | $0$ |
$144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
$120$ | $6$ | $(1,2,3,4,5,6)$ | $1$ |
$120$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |