Basic invariants
Dimension: | $15$ |
Group: | $S_7$ |
Conductor: | \(567\!\cdots\!849\)\(\medspace = 389^{10} \cdot 967^{10} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.1.376163.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 42T411 |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $S_7$ |
Projective stem field: | Galois closure of 7.1.376163.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - x^{6} - x^{4} - x^{3} + x^{2} + x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$: \( x^{2} + 97x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( 21 a + 39 + \left(69 a + 61\right)\cdot 101 + \left(36 a + 2\right)\cdot 101^{2} + \left(35 a + 48\right)\cdot 101^{3} + \left(13 a + 73\right)\cdot 101^{4} +O(101^{5})\) |
$r_{ 2 }$ | $=$ | \( 87 a + 70 + \left(91 a + 9\right)\cdot 101 + \left(46 a + 40\right)\cdot 101^{2} + \left(76 a + 51\right)\cdot 101^{3} + \left(67 a + 84\right)\cdot 101^{4} +O(101^{5})\) |
$r_{ 3 }$ | $=$ | \( 50 + 33\cdot 101 + 58\cdot 101^{2} + 33\cdot 101^{3} + 70\cdot 101^{4} +O(101^{5})\) |
$r_{ 4 }$ | $=$ | \( 80 a + 22 + \left(31 a + 14\right)\cdot 101 + \left(64 a + 80\right)\cdot 101^{2} + \left(65 a + 51\right)\cdot 101^{3} + \left(87 a + 91\right)\cdot 101^{4} +O(101^{5})\) |
$r_{ 5 }$ | $=$ | \( 42 a + 21 + \left(70 a + 29\right)\cdot 101 + \left(88 a + 2\right)\cdot 101^{2} + \left(92 a + 15\right)\cdot 101^{3} + \left(43 a + 63\right)\cdot 101^{4} +O(101^{5})\) |
$r_{ 6 }$ | $=$ | \( 14 a + 14 + \left(9 a + 88\right)\cdot 101 + \left(54 a + 34\right)\cdot 101^{2} + \left(24 a + 7\right)\cdot 101^{3} + \left(33 a + 77\right)\cdot 101^{4} +O(101^{5})\) |
$r_{ 7 }$ | $=$ | \( 59 a + 88 + \left(30 a + 66\right)\cdot 101 + \left(12 a + 84\right)\cdot 101^{2} + \left(8 a + 95\right)\cdot 101^{3} + \left(57 a + 44\right)\cdot 101^{4} +O(101^{5})\) |
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 |
$1$ | $1$ | $()$ | $15$ |
$21$ | $2$ | $(1,2)$ | $-5$ |
$105$ | $2$ | $(1,2)(3,4)(5,6)$ | $3$ |
$105$ | $2$ | $(1,2)(3,4)$ | $-1$ |
$70$ | $3$ | $(1,2,3)$ | $3$ |
$280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
$210$ | $4$ | $(1,2,3,4)$ | $-1$ |
$630$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
$504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
$210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
$420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
$840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
$720$ | $7$ | $(1,2,3,4,5,6,7)$ | $1$ |
$504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
$420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.