Basic invariants
Dimension: | $14$ |
Group: | $S_7$ |
Conductor: | \(833\!\cdots\!431\)\(\medspace = 7^{9} \cdot 38953^{9} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.1.272671.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 30T565 |
Parity: | odd |
Determinant: | 1.272671.2t1.a.a |
Projective image: | $S_7$ |
Projective stem field: | Galois closure of 7.1.272671.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - 2x^{6} + 3x^{5} - 4x^{4} + 4x^{3} - 3x^{2} + x - 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$: \( x^{2} + 69x + 7 \)
Roots:
$r_{ 1 }$ | $=$ | \( 23 a + 54 + \left(64 a + 36\right)\cdot 71 + \left(6 a + 68\right)\cdot 71^{2} + \left(5 a + 16\right)\cdot 71^{3} + \left(41 a + 27\right)\cdot 71^{4} +O(71^{5})\) |
$r_{ 2 }$ | $=$ | \( 27 a + 69 + \left(55 a + 31\right)\cdot 71 + \left(19 a + 34\right)\cdot 71^{2} + \left(15 a + 10\right)\cdot 71^{3} + \left(52 a + 26\right)\cdot 71^{4} +O(71^{5})\) |
$r_{ 3 }$ | $=$ | \( 58 + 9\cdot 71 + 60\cdot 71^{2} + 7\cdot 71^{3} + 39\cdot 71^{4} +O(71^{5})\) |
$r_{ 4 }$ | $=$ | \( 56 + 25\cdot 71 + 32\cdot 71^{2} + 12\cdot 71^{3} + 58\cdot 71^{4} +O(71^{5})\) |
$r_{ 5 }$ | $=$ | \( 48 a + 29 + 6 a\cdot 71 + \left(64 a + 18\right)\cdot 71^{2} + \left(65 a + 20\right)\cdot 71^{3} + \left(29 a + 33\right)\cdot 71^{4} +O(71^{5})\) |
$r_{ 6 }$ | $=$ | \( 44 a + 52 + \left(15 a + 44\right)\cdot 71 + \left(51 a + 18\right)\cdot 71^{2} + \left(55 a + 21\right)\cdot 71^{3} + \left(18 a + 44\right)\cdot 71^{4} +O(71^{5})\) |
$r_{ 7 }$ | $=$ | \( 39 + 63\cdot 71 + 51\cdot 71^{2} + 52\cdot 71^{3} + 55\cdot 71^{4} +O(71^{5})\) |
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 |
$1$ | $1$ | $()$ | $14$ |
$21$ | $2$ | $(1,2)$ | $-4$ |
$105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
$105$ | $2$ | $(1,2)(3,4)$ | $2$ |
$70$ | $3$ | $(1,2,3)$ | $-1$ |
$280$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
$210$ | $4$ | $(1,2,3,4)$ | $2$ |
$630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
$504$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
$210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
$420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
$840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
$720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
$504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $1$ |
$420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.