Basic invariants
Dimension: | $14$ |
Group: | $S_7$ |
Conductor: | \(537\!\cdots\!707\)\(\medspace = 351587^{5} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.1.351587.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 30T565 |
Parity: | odd |
Determinant: | 1.351587.2t1.a.a |
Projective image: | $S_7$ |
Projective stem field: | Galois closure of 7.1.351587.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - 2x^{5} - 2x^{4} + 2x^{3} + 3x^{2} - 1 \) . |
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:
$r_{ 1 }$ | $=$ | \( 4 a + 9 + \left(34 a + 4\right)\cdot 47 + \left(6 a + 38\right)\cdot 47^{2} + \left(29 a + 29\right)\cdot 47^{3} + \left(24 a + 12\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 2 }$ | $=$ | \( 6 a + 32 + \left(32 a + 42\right)\cdot 47 + \left(20 a + 18\right)\cdot 47^{2} + \left(28 a + 33\right)\cdot 47^{3} + \left(20 a + 28\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 3 }$ | $=$ | \( 41 a + 44 + \left(14 a + 6\right)\cdot 47 + \left(26 a + 28\right)\cdot 47^{2} + \left(18 a + 22\right)\cdot 47^{3} + \left(26 a + 41\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 4 }$ | $=$ | \( 22 a + 8 + \left(46 a + 24\right)\cdot 47 + \left(32 a + 35\right)\cdot 47^{2} + \left(8 a + 3\right)\cdot 47^{3} + \left(40 a + 34\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 5 }$ | $=$ | \( 43 a + 17 + \left(12 a + 21\right)\cdot 47 + \left(40 a + 17\right)\cdot 47^{2} + \left(17 a + 34\right)\cdot 47^{3} + \left(22 a + 32\right)\cdot 47^{4} +O(47^{5})\) |
$r_{ 6 }$ | $=$ | \( 26 + 40\cdot 47 + 41\cdot 47^{2} + 28\cdot 47^{3} + 26\cdot 47^{4} +O(47^{5})\) |
$r_{ 7 }$ | $=$ | \( 25 a + 5 + 47 + \left(14 a + 8\right)\cdot 47^{2} + \left(38 a + 35\right)\cdot 47^{3} + \left(6 a + 11\right)\cdot 47^{4} +O(47^{5})\) |
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.