Basic invariants
Dimension: | $6$ |
Group: | $S_7$ |
Conductor: | \(397991\)\(\medspace = 11 \cdot 97 \cdot 373 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.1.397991.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_7$ |
Parity: | odd |
Determinant: | 1.397991.2t1.a.a |
Projective image: | $S_7$ |
Projective stem field: | Galois closure of 7.1.397991.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - 2x^{5} + 3x^{3} - x^{2} - x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 83 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 83 }$: \( x^{2} + 82x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( 2 + 32\cdot 83 + 6\cdot 83^{2} + 41\cdot 83^{3} + 26\cdot 83^{4} +O(83^{5})\) |
$r_{ 2 }$ | $=$ | \( 37 + 31\cdot 83 + 6\cdot 83^{2} + 37\cdot 83^{3} + 10\cdot 83^{4} +O(83^{5})\) |
$r_{ 3 }$ | $=$ | \( 79 a + 76 + 32\cdot 83 + \left(52 a + 63\right)\cdot 83^{2} + \left(27 a + 23\right)\cdot 83^{3} + \left(63 a + 16\right)\cdot 83^{4} +O(83^{5})\) |
$r_{ 4 }$ | $=$ | \( 4 a + 72 + \left(82 a + 37\right)\cdot 83 + \left(30 a + 31\right)\cdot 83^{2} + \left(55 a + 82\right)\cdot 83^{3} + \left(19 a + 51\right)\cdot 83^{4} +O(83^{5})\) |
$r_{ 5 }$ | $=$ | \( 8 a + 76 + \left(51 a + 65\right)\cdot 83 + \left(45 a + 24\right)\cdot 83^{2} + \left(52 a + 71\right)\cdot 83^{3} + \left(18 a + 4\right)\cdot 83^{4} +O(83^{5})\) |
$r_{ 6 }$ | $=$ | \( 68 + 22\cdot 83 + 14\cdot 83^{2} + 81\cdot 83^{3} + 83^{4} +O(83^{5})\) |
$r_{ 7 }$ | $=$ | \( 75 a + 1 + \left(31 a + 26\right)\cdot 83 + \left(37 a + 19\right)\cdot 83^{2} + \left(30 a + 78\right)\cdot 83^{3} + \left(64 a + 53\right)\cdot 83^{4} +O(83^{5})\) |
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.