Basic invariants
Dimension: | $14$ |
Group: | $S_7$ |
Conductor: | \(168\!\cdots\!281\)\(\medspace = 202471^{4} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.1.202471.1 |
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 7.1.202471.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - x^{6} - x^{5} + 2x^{4} - x^{2} + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: \( x^{2} + 29x + 3 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 8 + 2\cdot 31 + 16\cdot 31^{2} + 19\cdot 31^{3} + 14\cdot 31^{4} +O(31^{5})\)
$r_{ 2 }$ |
$=$ |
\( 12 a + 3 + \left(a + 2\right)\cdot 31 + 30 a\cdot 31^{2} + \left(25 a + 19\right)\cdot 31^{3} + \left(8 a + 23\right)\cdot 31^{4} +O(31^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 4 a + 3 + \left(17 a + 25\right)\cdot 31 + \left(30 a + 4\right)\cdot 31^{2} + \left(18 a + 12\right)\cdot 31^{3} + \left(8 a + 25\right)\cdot 31^{4} +O(31^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 27 a + 11 + \left(13 a + 24\right)\cdot 31 + 17\cdot 31^{2} + \left(12 a + 19\right)\cdot 31^{3} + \left(22 a + 23\right)\cdot 31^{4} +O(31^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 9 a + 12 + \left(30 a + 28\right)\cdot 31 + \left(24 a + 18\right)\cdot 31^{2} + \left(22 a + 11\right)\cdot 31^{3} + \left(16 a + 5\right)\cdot 31^{4} +O(31^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 19 a + 27 + \left(29 a + 23\right)\cdot 31 + 27\cdot 31^{2} + \left(5 a + 9\right)\cdot 31^{3} + \left(22 a + 15\right)\cdot 31^{4} +O(31^{5})\)
| $r_{ 7 }$ |
$=$ |
\( 22 a + 30 + 17\cdot 31 + \left(6 a + 7\right)\cdot 31^{2} + \left(8 a + 1\right)\cdot 31^{3} + \left(14 a + 16\right)\cdot 31^{4} +O(31^{5})\)
| |
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.