Basic invariants
Dimension: | $14$ |
Group: | $S_7$ |
Conductor: | \(705\!\cdots\!721\)\(\medspace = 109^{4} \cdot 2659^{4} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.1.289831.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 21T38 |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $S_7$ |
Projective stem field: | Galois closure of 7.1.289831.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} + x^{5} - x^{4} + x^{3} - x^{2} + x - 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 83 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 83 }$: \( x^{2} + 82x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( 81 a + 20 + \left(81 a + 80\right)\cdot 83 + \left(18 a + 17\right)\cdot 83^{2} + \left(44 a + 62\right)\cdot 83^{3} + \left(69 a + 17\right)\cdot 83^{4} +O(83^{5})\) |
$r_{ 2 }$ | $=$ | \( 2 a + 40 + \left(41 a + 10\right)\cdot 83 + \left(43 a + 69\right)\cdot 83^{2} + \left(3 a + 66\right)\cdot 83^{3} + \left(14 a + 64\right)\cdot 83^{4} +O(83^{5})\) |
$r_{ 3 }$ | $=$ | \( 27 a + 30 + \left(21 a + 23\right)\cdot 83 + \left(2 a + 43\right)\cdot 83^{2} + \left(29 a + 8\right)\cdot 83^{3} + \left(70 a + 62\right)\cdot 83^{4} +O(83^{5})\) |
$r_{ 4 }$ | $=$ | \( 2 a + 18 + \left(a + 81\right)\cdot 83 + \left(64 a + 37\right)\cdot 83^{2} + \left(38 a + 4\right)\cdot 83^{3} + \left(13 a + 43\right)\cdot 83^{4} +O(83^{5})\) |
$r_{ 5 }$ | $=$ | \( 56 a + 57 + \left(61 a + 17\right)\cdot 83 + \left(80 a + 24\right)\cdot 83^{2} + \left(53 a + 35\right)\cdot 83^{3} + \left(12 a + 20\right)\cdot 83^{4} +O(83^{5})\) |
$r_{ 6 }$ | $=$ | \( 81 a + 42 + \left(41 a + 49\right)\cdot 83 + \left(39 a + 71\right)\cdot 83^{2} + \left(79 a + 26\right)\cdot 83^{3} + \left(68 a + 75\right)\cdot 83^{4} +O(83^{5})\) |
$r_{ 7 }$ | $=$ | \( 42 + 69\cdot 83 + 67\cdot 83^{2} + 44\cdot 83^{3} + 48\cdot 83^{4} +O(83^{5})\) |
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 |
$1$ | $1$ | $()$ | $14$ |
$21$ | $2$ | $(1,2)$ | $6$ |
$105$ | $2$ | $(1,2)(3,4)(5,6)$ | $2$ |
$105$ | $2$ | $(1,2)(3,4)$ | $2$ |
$70$ | $3$ | $(1,2,3)$ | $2$ |
$280$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
$210$ | $4$ | $(1,2,3,4)$ | $0$ |
$630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
$504$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
$210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $2$ |
$420$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
$840$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
$720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
$504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $1$ |
$420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.