Basic invariants
Dimension: | $20$ |
Group: | $S_7$ |
Conductor: | \(954\!\cdots\!169\)\(\medspace = 3^{24} \cdot 7127^{10} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.1.577287.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 70 |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $S_7$ |
Projective stem field: | Galois closure of 7.1.577287.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - 2x^{6} + x^{5} + x^{2} - x - 1 \) . |
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:
$r_{ 1 }$ | $=$ | \( 39 a + 28 + \left(16 a + 38\right)\cdot 47 + \left(5 a + 35\right)\cdot 47^{2} + \left(30 a + 20\right)\cdot 47^{3} + \left(45 a + 10\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 2 }$ | $=$ | \( 42 a + 26 + \left(35 a + 5\right)\cdot 47 + \left(9 a + 9\right)\cdot 47^{2} + \left(31 a + 44\right)\cdot 47^{3} + \left(4 a + 40\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 3 }$ | $=$ | \( 27 + 18\cdot 47 + 44\cdot 47^{2} + 46\cdot 47^{3} + 8\cdot 47^{4} +O(47^{5})\) |
$r_{ 4 }$ | $=$ | \( 5 a + 16 + \left(11 a + 35\right)\cdot 47 + \left(37 a + 39\right)\cdot 47^{2} + \left(15 a + 2\right)\cdot 47^{3} + \left(42 a + 19\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 5 }$ | $=$ | \( 42 + 30\cdot 47 + 46\cdot 47^{2} + 38\cdot 47^{3} + 45\cdot 47^{4} +O(47^{5})\) |
$r_{ 6 }$ | $=$ | \( 39 + 25\cdot 47 + 29\cdot 47^{2} + 5\cdot 47^{3} + 38\cdot 47^{4} +O(47^{5})\) |
$r_{ 7 }$ | $=$ | \( 8 a + 12 + \left(30 a + 33\right)\cdot 47 + \left(41 a + 29\right)\cdot 47^{2} + \left(16 a + 28\right)\cdot 47^{3} + \left(a + 24\right)\cdot 47^{4} +O(47^{5})\) |
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 |
$1$ | $1$ | $()$ | $20$ |
$21$ | $2$ | $(1,2)$ | $0$ |
$105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
$105$ | $2$ | $(1,2)(3,4)$ | $-4$ |
$70$ | $3$ | $(1,2,3)$ | $2$ |
$280$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
$210$ | $4$ | $(1,2,3,4)$ | $0$ |
$630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
$504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
$210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $2$ |
$420$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
$840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
$720$ | $7$ | $(1,2,3,4,5,6,7)$ | $-1$ |
$504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
$420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.