Basic invariants
Dimension: | $6$ |
Group: | $S_7$ |
Conductor: | \(311071\)\(\medspace = 277 \cdot 1123 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.1.311071.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_7$ |
Parity: | odd |
Determinant: | 1.311071.2t1.a.a |
Projective image: | $S_7$ |
Projective stem field: | Galois closure of 7.1.311071.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - x^{6} + 3x^{5} - 3x^{4} + 4x^{3} - 4x^{2} + 2x - 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$: \( x^{2} + 96x + 5 \)
Roots:
$r_{ 1 }$ | $=$ | \( 8 + 75\cdot 97 + 45\cdot 97^{2} + 83\cdot 97^{3} + 21\cdot 97^{4} +O(97^{5})\) |
$r_{ 2 }$ | $=$ | \( 54 a + 47 + \left(16 a + 94\right)\cdot 97 + \left(9 a + 77\right)\cdot 97^{2} + \left(54 a + 62\right)\cdot 97^{3} + \left(20 a + 93\right)\cdot 97^{4} +O(97^{5})\) |
$r_{ 3 }$ | $=$ | \( 43 a + 4 + \left(80 a + 57\right)\cdot 97 + \left(87 a + 70\right)\cdot 97^{2} + \left(42 a + 10\right)\cdot 97^{3} + \left(76 a + 60\right)\cdot 97^{4} +O(97^{5})\) |
$r_{ 4 }$ | $=$ | \( 87 a + 25 + \left(64 a + 27\right)\cdot 97 + \left(4 a + 58\right)\cdot 97^{2} + \left(29 a + 58\right)\cdot 97^{3} + \left(53 a + 63\right)\cdot 97^{4} +O(97^{5})\) |
$r_{ 5 }$ | $=$ | \( 10 a + 15 + \left(32 a + 5\right)\cdot 97 + \left(92 a + 95\right)\cdot 97^{2} + \left(67 a + 82\right)\cdot 97^{3} + \left(43 a + 87\right)\cdot 97^{4} +O(97^{5})\) |
$r_{ 6 }$ | $=$ | \( 41 + 97 + 15\cdot 97^{2} + 97^{3} + 41\cdot 97^{4} +O(97^{5})\) |
$r_{ 7 }$ | $=$ | \( 55 + 30\cdot 97 + 25\cdot 97^{2} + 88\cdot 97^{3} + 19\cdot 97^{4} +O(97^{5})\) |
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.