Basic invariants
Dimension: | $6$ |
Group: | $S_7$ |
Conductor: | \(315631\) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.1.315631.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_7$ |
Parity: | odd |
Determinant: | 1.315631.2t1.a.a |
Projective image: | $S_7$ |
Projective stem field: | Galois closure of 7.1.315631.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - x^{5} - 2x^{4} + x^{2} + x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 163 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 163 }$: \( x^{2} + 159x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( 46 + 120\cdot 163 + 63\cdot 163^{2} + 62\cdot 163^{3} + 5\cdot 163^{4} +O(163^{5})\) |
$r_{ 2 }$ | $=$ | \( 100 a + 25 + \left(71 a + 81\right)\cdot 163 + \left(39 a + 131\right)\cdot 163^{2} + \left(2 a + 129\right)\cdot 163^{3} + \left(66 a + 60\right)\cdot 163^{4} +O(163^{5})\) |
$r_{ 3 }$ | $=$ | \( 38 a + 23 + \left(162 a + 69\right)\cdot 163 + \left(76 a + 150\right)\cdot 163^{2} + \left(49 a + 65\right)\cdot 163^{3} + \left(132 a + 143\right)\cdot 163^{4} +O(163^{5})\) |
$r_{ 4 }$ | $=$ | \( 109 + 86\cdot 163 + 9\cdot 163^{2} + 100\cdot 163^{3} + 126\cdot 163^{4} +O(163^{5})\) |
$r_{ 5 }$ | $=$ | \( 12 + 162\cdot 163 + 108\cdot 163^{2} + 7\cdot 163^{3} + 22\cdot 163^{4} +O(163^{5})\) |
$r_{ 6 }$ | $=$ | \( 125 a + 12 + 28\cdot 163 + \left(86 a + 133\right)\cdot 163^{2} + \left(113 a + 23\right)\cdot 163^{3} + \left(30 a + 134\right)\cdot 163^{4} +O(163^{5})\) |
$r_{ 7 }$ | $=$ | \( 63 a + 99 + \left(91 a + 104\right)\cdot 163 + \left(123 a + 54\right)\cdot 163^{2} + \left(160 a + 99\right)\cdot 163^{3} + \left(96 a + 159\right)\cdot 163^{4} +O(163^{5})\) |
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.