Basic invariants
Dimension: | $6$ |
Group: | $S_7$ |
Conductor: | \(320975\)\(\medspace = 5^{2} \cdot 37 \cdot 347 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.1.320975.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_7$ |
Parity: | odd |
Determinant: | 1.12839.2t1.a.a |
Projective image: | $S_7$ |
Projective stem field: | Galois closure of 7.1.320975.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - x^{6} + x^{5} - x^{4} + 2x^{2} - 2x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: \( x^{2} + 49x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( 44 a + 35 + \left(25 a + 4\right)\cdot 53 + \left(49 a + 10\right)\cdot 53^{2} + \left(11 a + 34\right)\cdot 53^{3} + 35\cdot 53^{4} +O(53^{5})\) |
$r_{ 2 }$ | $=$ | \( 14 + 21\cdot 53 + 16\cdot 53^{2} + 29\cdot 53^{3} + 35\cdot 53^{4} +O(53^{5})\) |
$r_{ 3 }$ | $=$ | \( 9 a + 52 + \left(27 a + 10\right)\cdot 53 + \left(3 a + 23\right)\cdot 53^{2} + \left(41 a + 32\right)\cdot 53^{3} + \left(52 a + 24\right)\cdot 53^{4} +O(53^{5})\) |
$r_{ 4 }$ | $=$ | \( 52 a + 51 + \left(41 a + 51\right)\cdot 53 + \left(20 a + 39\right)\cdot 53^{2} + \left(31 a + 47\right)\cdot 53^{3} + \left(25 a + 16\right)\cdot 53^{4} +O(53^{5})\) |
$r_{ 5 }$ | $=$ | \( 31 + 2\cdot 53 + 52\cdot 53^{2} + 10\cdot 53^{3} + 26\cdot 53^{4} +O(53^{5})\) |
$r_{ 6 }$ | $=$ | \( a + 47 + \left(11 a + 8\right)\cdot 53 + \left(32 a + 28\right)\cdot 53^{2} + \left(21 a + 46\right)\cdot 53^{3} + \left(27 a + 34\right)\cdot 53^{4} +O(53^{5})\) |
$r_{ 7 }$ | $=$ | \( 36 + 5\cdot 53 + 42\cdot 53^{2} + 10\cdot 53^{3} + 38\cdot 53^{4} +O(53^{5})\) |
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.