Basic invariants
Dimension: | $6$ |
Group: | $S_7$ |
Conductor: | \(380831\)\(\medspace = 11 \cdot 89 \cdot 389 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.1.380831.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_7$ |
Parity: | odd |
Determinant: | 1.380831.2t1.a.a |
Projective image: | $S_7$ |
Projective stem field: | Galois closure of 7.1.380831.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - x^{5} - x^{4} - x^{3} + 2x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: \( x^{2} + 42x + 3 \)
Roots:
$r_{ 1 }$ | $=$ | \( 22 a + 16 + \left(10 a + 30\right)\cdot 43 + \left(38 a + 23\right)\cdot 43^{2} + 16\cdot 43^{3} + \left(3 a + 24\right)\cdot 43^{4} +O(43^{5})\) |
$r_{ 2 }$ | $=$ | \( 32 a + 16 + \left(7 a + 36\right)\cdot 43 + \left(24 a + 36\right)\cdot 43^{2} + \left(33 a + 16\right)\cdot 43^{3} + \left(22 a + 17\right)\cdot 43^{4} +O(43^{5})\) |
$r_{ 3 }$ | $=$ | \( 11 a + 5 + \left(35 a + 12\right)\cdot 43 + \left(18 a + 10\right)\cdot 43^{2} + \left(9 a + 26\right)\cdot 43^{3} + \left(20 a + 6\right)\cdot 43^{4} +O(43^{5})\) |
$r_{ 4 }$ | $=$ | \( 5 + 8\cdot 43 + 14\cdot 43^{2} + 10\cdot 43^{3} +O(43^{5})\) |
$r_{ 5 }$ | $=$ | \( 6 a + \left(4 a + 34\right)\cdot 43 + \left(25 a + 28\right)\cdot 43^{2} + \left(29 a + 37\right)\cdot 43^{3} + \left(23 a + 29\right)\cdot 43^{4} +O(43^{5})\) |
$r_{ 6 }$ | $=$ | \( 37 a + 6 + \left(38 a + 32\right)\cdot 43 + \left(17 a + 6\right)\cdot 43^{2} + \left(13 a + 42\right)\cdot 43^{3} + \left(19 a + 23\right)\cdot 43^{4} +O(43^{5})\) |
$r_{ 7 }$ | $=$ | \( 21 a + 38 + \left(32 a + 18\right)\cdot 43 + \left(4 a + 8\right)\cdot 43^{2} + \left(42 a + 22\right)\cdot 43^{3} + \left(39 a + 26\right)\cdot 43^{4} +O(43^{5})\) |
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 |
$1$ | $1$ | $()$ | $6$ |
$21$ | $2$ | $(1,2)$ | $4$ |
$105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
$105$ | $2$ | $(1,2)(3,4)$ | $2$ |
$70$ | $3$ | $(1,2,3)$ | $3$ |
$280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
$210$ | $4$ | $(1,2,3,4)$ | $2$ |
$630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
$504$ | $5$ | $(1,2,3,4,5)$ | $1$ |
$210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
$420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
$840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
$720$ | $7$ | $(1,2,3,4,5,6,7)$ | $-1$ |
$504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $-1$ |
$420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.