Basic invariants
Dimension: | $6$ |
Group: | $S_7$ |
Conductor: | \(317159\) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 7.1.317159.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_7$ |
Parity: | odd |
Projective image: | $S_7$ |
Projective field: | Galois closure of 7.1.317159.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$:
\( x^{2} + 45x + 5 \)
Roots:
$r_{ 1 }$ | $=$ | \( 34 a + 14 + \left(12 a + 39\right)\cdot 47 + \left(23 a + 3\right)\cdot 47^{2} + \left(22 a + 2\right)\cdot 47^{3} + \left(43 a + 46\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 2 }$ | $=$ | \( 20 a + 9 + \left(17 a + 26\right)\cdot 47 + \left(26 a + 19\right)\cdot 47^{2} + \left(27 a + 2\right)\cdot 47^{3} + \left(5 a + 37\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 3 }$ | $=$ | \( 11 a + 23 + \left(30 a + 21\right)\cdot 47 + \left(9 a + 4\right)\cdot 47^{2} + \left(7 a + 43\right)\cdot 47^{3} + \left(34 a + 21\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 4 }$ | $=$ | \( 36 a + 45 + \left(16 a + 23\right)\cdot 47 + \left(37 a + 40\right)\cdot 47^{2} + 39 a\cdot 47^{3} + \left(12 a + 36\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 5 }$ | $=$ | \( 16 + 5\cdot 47 + 27\cdot 47^{2} + 37\cdot 47^{3} + 9\cdot 47^{4} +O(47^{5})\) |
$r_{ 6 }$ | $=$ | \( 27 a + 2 + \left(29 a + 41\right)\cdot 47 + \left(20 a + 7\right)\cdot 47^{2} + \left(19 a + 31\right)\cdot 47^{3} + \left(41 a + 20\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 7 }$ | $=$ | \( 13 a + 35 + \left(34 a + 30\right)\cdot 47 + \left(23 a + 37\right)\cdot 47^{2} + \left(24 a + 23\right)\cdot 47^{3} + \left(3 a + 16\right)\cdot 47^{4} +O(47^{5})\) |
Generators of the action on the roots $r_1, \ldots, r_{ 7 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 7 }$ | Character values |
$c1$ | |||
$1$ | $1$ | $()$ | $6$ |
$21$ | $2$ | $(1,2)$ | $4$ |
$105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
$105$ | $2$ | $(1,2)(3,4)$ | $2$ |
$70$ | $3$ | $(1,2,3)$ | $3$ |
$280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
$210$ | $4$ | $(1,2,3,4)$ | $2$ |
$630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
$504$ | $5$ | $(1,2,3,4,5)$ | $1$ |
$210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
$420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
$840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
$720$ | $7$ | $(1,2,3,4,5,6,7)$ | $-1$ |
$504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $-1$ |
$420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |