Basic invariants
Dimension: | $6$ |
Group: | $S_7$ |
Conductor: | \(535567\)\(\medspace = 139 \cdot 3853 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.1.535567.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_7$ |
Parity: | odd |
Determinant: | 1.535567.2t1.a.a |
Projective image: | $S_7$ |
Projective stem field: | Galois closure of 7.1.535567.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - x^{6} + x^{5} - 2x^{4} + 2x^{3} - x^{2} + 2x - 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: \( x^{2} + 21x + 5 \)
Roots:
$r_{ 1 }$ | $=$ | \( 19 a + 11 + \left(13 a + 3\right)\cdot 23 + \left(a + 2\right)\cdot 23^{2} + \left(20 a + 11\right)\cdot 23^{3} + \left(12 a + 9\right)\cdot 23^{4} +O(23^{5})\) |
$r_{ 2 }$ | $=$ | \( 8 a + 5 + \left(10 a + 18\right)\cdot 23 + \left(2 a + 20\right)\cdot 23^{2} + \left(4 a + 10\right)\cdot 23^{3} + \left(18 a + 20\right)\cdot 23^{4} +O(23^{5})\) |
$r_{ 3 }$ | $=$ | \( 15 a + 21 + \left(12 a + 7\right)\cdot 23 + \left(20 a + 15\right)\cdot 23^{2} + \left(18 a + 16\right)\cdot 23^{3} + \left(4 a + 6\right)\cdot 23^{4} +O(23^{5})\) |
$r_{ 4 }$ | $=$ | \( a + 18 + \left(7 a + 1\right)\cdot 23 + \left(5 a + 14\right)\cdot 23^{2} + \left(17 a + 5\right)\cdot 23^{3} + \left(17 a + 9\right)\cdot 23^{4} +O(23^{5})\) |
$r_{ 5 }$ | $=$ | \( 4 a + 3 + \left(9 a + 12\right)\cdot 23 + \left(21 a + 14\right)\cdot 23^{2} + \left(2 a + 3\right)\cdot 23^{3} + \left(10 a + 15\right)\cdot 23^{4} +O(23^{5})\) |
$r_{ 6 }$ | $=$ | \( 15 + 10\cdot 23 + 7\cdot 23^{2} + 9\cdot 23^{3} + 3\cdot 23^{4} +O(23^{5})\) |
$r_{ 7 }$ | $=$ | \( 22 a + 20 + \left(15 a + 14\right)\cdot 23 + \left(17 a + 17\right)\cdot 23^{2} + \left(5 a + 11\right)\cdot 23^{3} + \left(5 a + 4\right)\cdot 23^{4} +O(23^{5})\) |
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.