Basic invariants
Dimension: | $14$ |
Group: | $S_7$ |
Conductor: | \(318\!\cdots\!193\)\(\medspace = 126039593^{5} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.7.126039593.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 30T565 |
Parity: | even |
Determinant: | 1.126039593.2t1.a.a |
Projective image: | $S_7$ |
Projective stem field: | Galois closure of 7.7.126039593.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - x^{6} - 7x^{5} + 5x^{4} + 14x^{3} - 7x^{2} - 7x + 3 \) . |
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:
$r_{ 1 }$ | $=$ | \( 20 a + 16 + \left(46 a + 5\right)\cdot 47 + \left(15 a + 32\right)\cdot 47^{2} + \left(43 a + 40\right)\cdot 47^{3} + \left(18 a + 3\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 2 }$ | $=$ | \( 7 a + 20 + \left(28 a + 2\right)\cdot 47 + \left(17 a + 44\right)\cdot 47^{2} + \left(36 a + 16\right)\cdot 47^{3} + \left(41 a + 6\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 3 }$ | $=$ | \( 27 a + 9 + 31\cdot 47 + \left(31 a + 17\right)\cdot 47^{2} + \left(3 a + 17\right)\cdot 47^{3} + \left(28 a + 45\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 4 }$ | $=$ | \( 15 + 41\cdot 47 + 16\cdot 47^{2} + 26\cdot 47^{3} + 24\cdot 47^{4} +O(47^{5})\) |
$r_{ 5 }$ | $=$ | \( 40 a + 34 + \left(18 a + 4\right)\cdot 47 + \left(29 a + 4\right)\cdot 47^{2} + \left(10 a + 25\right)\cdot 47^{3} + \left(5 a + 6\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 6 }$ | $=$ | \( 43 a + 28 + \left(2 a + 46\right)\cdot 47 + \left(12 a + 25\right)\cdot 47^{2} + \left(11 a + 25\right)\cdot 47^{3} + \left(5 a + 27\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 7 }$ | $=$ | \( 4 a + 20 + \left(44 a + 9\right)\cdot 47 + 34 a\cdot 47^{2} + \left(35 a + 36\right)\cdot 47^{3} + \left(41 a + 26\right)\cdot 47^{4} +O(47^{5})\) |
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 value |
$1$ | $1$ | $()$ | $14$ |
$21$ | $2$ | $(1,2)$ | $4$ |
$105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
$105$ | $2$ | $(1,2)(3,4)$ | $2$ |
$70$ | $3$ | $(1,2,3)$ | $-1$ |
$280$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
$210$ | $4$ | $(1,2,3,4)$ | $-2$ |
$630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
$504$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
$210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
$420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
$840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
$720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
$504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $-1$ |
$420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.