Basic invariants
Dimension: | $6$ |
Group: | $S_7$ |
Conductor: | \(227287\)\(\medspace = 167 \cdot 1361 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.1.227287.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_7$ |
Parity: | odd |
Determinant: | 1.227287.2t1.a.a |
Projective image: | $S_7$ |
Projective stem field: | Galois closure of 7.1.227287.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - 2x^{6} + 3x^{5} - 3x^{4} + 3x^{3} - 2x^{2} + 2x - 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: \( x^{2} + 18x + 2 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 17 a + 16 + \left(17 a + 8\right)\cdot 19 + \left(11 a + 6\right)\cdot 19^{2} + \left(7 a + 18\right)\cdot 19^{3} + \left(14 a + 5\right)\cdot 19^{4} +O(19^{5})\)
$r_{ 2 }$ |
$=$ |
\( 17 + 16\cdot 19 + 16\cdot 19^{2} + 19^{3} + 13\cdot 19^{4} +O(19^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 5 a + 1 + \left(3 a + 10\right)\cdot 19 + \left(17 a + 1\right)\cdot 19^{2} + \left(3 a + 15\right)\cdot 19^{3} + \left(18 a + 14\right)\cdot 19^{4} +O(19^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 2 a + 14 + \left(a + 9\right)\cdot 19 + 7 a\cdot 19^{2} + \left(11 a + 14\right)\cdot 19^{3} + \left(4 a + 12\right)\cdot 19^{4} +O(19^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 5 a + \left(10 a + 18\right)\cdot 19 + \left(15 a + 14\right)\cdot 19^{2} + \left(18 a + 10\right)\cdot 19^{3} + \left(6 a + 15\right)\cdot 19^{4} +O(19^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 14 a + 6 + \left(15 a + 8\right)\cdot 19 + \left(a + 15\right)\cdot 19^{2} + \left(15 a + 1\right)\cdot 19^{3} + 10\cdot 19^{4} +O(19^{5})\)
| $r_{ 7 }$ |
$=$ |
\( 14 a + 5 + \left(8 a + 4\right)\cdot 19 + \left(3 a + 1\right)\cdot 19^{2} + 14\cdot 19^{3} + \left(12 a + 3\right)\cdot 19^{4} +O(19^{5})\)
| |
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.