Basic invariants
Dimension: | $6$ |
Group: | $S_7$ |
Conductor: | \(349847\)\(\medspace = 19 \cdot 18413 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.1.349847.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_7$ |
Parity: | odd |
Determinant: | 1.349847.2t1.a.a |
Projective image: | $S_7$ |
Projective stem field: | Galois closure of 7.1.349847.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - x^{6} + x^{5} - x^{4} + x^{2} - x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 89 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 89 }$: \( x^{2} + 82x + 3 \)
Roots:
$r_{ 1 }$ | $=$ | \( 35 a + 73 + \left(13 a + 3\right)\cdot 89 + \left(49 a + 16\right)\cdot 89^{2} + \left(2 a + 43\right)\cdot 89^{3} + \left(16 a + 58\right)\cdot 89^{4} +O(89^{5})\) |
$r_{ 2 }$ | $=$ | \( 67 + 28\cdot 89 + 19\cdot 89^{2} + 84\cdot 89^{3} + 19\cdot 89^{4} +O(89^{5})\) |
$r_{ 3 }$ | $=$ | \( 53 a + 10 + \left(41 a + 26\right)\cdot 89 + \left(2 a + 81\right)\cdot 89^{2} + \left(51 a + 20\right)\cdot 89^{3} + \left(2 a + 47\right)\cdot 89^{4} +O(89^{5})\) |
$r_{ 4 }$ | $=$ | \( 36 a + 25 + \left(47 a + 86\right)\cdot 89 + \left(86 a + 56\right)\cdot 89^{2} + \left(37 a + 19\right)\cdot 89^{3} + \left(86 a + 14\right)\cdot 89^{4} +O(89^{5})\) |
$r_{ 5 }$ | $=$ | \( 54 a + 51 + \left(75 a + 62\right)\cdot 89 + \left(39 a + 79\right)\cdot 89^{2} + \left(86 a + 11\right)\cdot 89^{3} + \left(72 a + 79\right)\cdot 89^{4} +O(89^{5})\) |
$r_{ 6 }$ | $=$ | \( 71 a + 84 + \left(71 a + 80\right)\cdot 89 + \left(31 a + 64\right)\cdot 89^{2} + \left(70 a + 35\right)\cdot 89^{3} + \left(31 a + 81\right)\cdot 89^{4} +O(89^{5})\) |
$r_{ 7 }$ | $=$ | \( 18 a + 47 + \left(17 a + 67\right)\cdot 89 + \left(57 a + 37\right)\cdot 89^{2} + \left(18 a + 51\right)\cdot 89^{3} + \left(57 a + 55\right)\cdot 89^{4} +O(89^{5})\) |
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.