Basic invariants
Dimension: | $14$ |
Group: | $S_7$ |
Conductor: | \(577\!\cdots\!001\)\(\medspace = 23^{10} \cdot 10337^{10} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.1.237751.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 42T413 |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $S_7$ |
Projective stem field: | Galois closure of 7.1.237751.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - x^{6} + x^{5} - x^{4} + x^{3} + x^{2} - 2x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: \( x^{2} + 38x + 6 \)
Roots:
$r_{ 1 }$ | $=$ | \( 30 a + 17 + \left(16 a + 20\right)\cdot 41 + \left(17 a + 22\right)\cdot 41^{2} + \left(14 a + 22\right)\cdot 41^{3} + \left(13 a + 8\right)\cdot 41^{4} +O(41^{5})\) |
$r_{ 2 }$ | $=$ | \( 34 a + 37 + \left(18 a + 38\right)\cdot 41 + \left(8 a + 25\right)\cdot 41^{2} + \left(17 a + 13\right)\cdot 41^{3} + \left(3 a + 29\right)\cdot 41^{4} +O(41^{5})\) |
$r_{ 3 }$ | $=$ | \( 35 + 31\cdot 41 + 28\cdot 41^{2} + 33\cdot 41^{3} + 6\cdot 41^{4} +O(41^{5})\) |
$r_{ 4 }$ | $=$ | \( 7 a + 16 + \left(22 a + 20\right)\cdot 41 + \left(32 a + 32\right)\cdot 41^{2} + \left(23 a + 15\right)\cdot 41^{3} + \left(37 a + 22\right)\cdot 41^{4} +O(41^{5})\) |
$r_{ 5 }$ | $=$ | \( 22 + 36\cdot 41 + 29\cdot 41^{2} + 14\cdot 41^{3} + 24\cdot 41^{4} +O(41^{5})\) |
$r_{ 6 }$ | $=$ | \( 13 + 16\cdot 41 + 7\cdot 41^{2} + 15\cdot 41^{3} + 38\cdot 41^{4} +O(41^{5})\) |
$r_{ 7 }$ | $=$ | \( 11 a + 25 + \left(24 a + 40\right)\cdot 41 + \left(23 a + 16\right)\cdot 41^{2} + \left(26 a + 7\right)\cdot 41^{3} + \left(27 a + 34\right)\cdot 41^{4} +O(41^{5})\) |
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 |
$1$ | $1$ | $()$ | $14$ |
$21$ | $2$ | $(1,2)$ | $-6$ |
$105$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
$105$ | $2$ | $(1,2)(3,4)$ | $2$ |
$70$ | $3$ | $(1,2,3)$ | $2$ |
$280$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
$210$ | $4$ | $(1,2,3,4)$ | $0$ |
$630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
$504$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
$210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $2$ |
$420$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
$840$ | $6$ | $(1,2,3,4,5,6)$ | $1$ |
$720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
$504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $-1$ |
$420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.