Basic invariants
Dimension: | $14$ |
Group: | $S_7$ |
Conductor: | \(147\!\cdots\!641\)\(\medspace = 29^{4} \cdot 6763^{4} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.1.196127.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 21T38 |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $S_7$ |
Projective stem field: | Galois closure of 7.1.196127.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - 2x^{6} + 2x^{5} - x^{4} + 2x^{2} - 2x + 1 \) . |
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:
$r_{ 1 }$ | $=$ | \( 6 + 33\cdot 41 + 25\cdot 41^{2} + 11\cdot 41^{3} + 38\cdot 41^{4} +O(41^{5})\) |
$r_{ 2 }$ | $=$ | \( 13 a + 36 + \left(17 a + 40\right)\cdot 41 + \left(19 a + 22\right)\cdot 41^{2} + \left(3 a + 3\right)\cdot 41^{3} + 20\cdot 41^{4} +O(41^{5})\) |
$r_{ 3 }$ | $=$ | \( 27 a + \left(5 a + 19\right)\cdot 41 + \left(29 a + 2\right)\cdot 41^{2} + \left(30 a + 21\right)\cdot 41^{3} + \left(20 a + 17\right)\cdot 41^{4} +O(41^{5})\) |
$r_{ 4 }$ | $=$ | \( 14 a + 40 + \left(35 a + 8\right)\cdot 41 + \left(11 a + 2\right)\cdot 41^{2} + \left(10 a + 2\right)\cdot 41^{3} + \left(20 a + 8\right)\cdot 41^{4} +O(41^{5})\) |
$r_{ 5 }$ | $=$ | \( 24 a + 30 + \left(9 a + 29\right)\cdot 41 + \left(4 a + 21\right)\cdot 41^{2} + \left(30 a + 22\right)\cdot 41^{3} + \left(3 a + 20\right)\cdot 41^{4} +O(41^{5})\) |
$r_{ 6 }$ | $=$ | \( 28 a + 34 + \left(23 a + 38\right)\cdot 41 + \left(21 a + 22\right)\cdot 41^{2} + \left(37 a + 35\right)\cdot 41^{3} + \left(40 a + 16\right)\cdot 41^{4} +O(41^{5})\) |
$r_{ 7 }$ | $=$ | \( 17 a + 20 + \left(31 a + 34\right)\cdot 41 + \left(36 a + 24\right)\cdot 41^{2} + \left(10 a + 26\right)\cdot 41^{3} + \left(37 a + 1\right)\cdot 41^{4} +O(41^{5})\) |
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.