Basic invariants
| Dimension: | $14$ |
| Group: | $S_7$ |
| Conductor: | \(989\!\cdots\!049\)\(\medspace = 499^{10} \cdot 797^{10} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.1.397703.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 42T413 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.1.397703.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} + x^{5} + x^{3} - 2x^{2} + x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$:
\( x^{2} + 58x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 35 a + 50 + \left(55 a + 29\right)\cdot 59 + \left(30 a + 14\right)\cdot 59^{2} + \left(23 a + 29\right)\cdot 59^{3} + \left(40 a + 30\right)\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 24 a + 26 + \left(3 a + 50\right)\cdot 59 + \left(28 a + 48\right)\cdot 59^{2} + \left(35 a + 21\right)\cdot 59^{3} + \left(18 a + 47\right)\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 16 a + 15 + \left(7 a + 38\right)\cdot 59 + \left(38 a + 6\right)\cdot 59^{2} + \left(54 a + 31\right)\cdot 59^{3} + \left(12 a + 11\right)\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 36 + 37\cdot 59 + 25\cdot 59^{3} + 47\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 13 + 29\cdot 59 + 52\cdot 59^{2} + 40\cdot 59^{3} + 16\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 43 a + 31 + \left(51 a + 29\right)\cdot 59 + \left(20 a + 37\right)\cdot 59^{2} + \left(4 a + 47\right)\cdot 59^{3} + \left(46 a + 28\right)\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 6 + 21\cdot 59 + 16\cdot 59^{2} + 40\cdot 59^{3} + 53\cdot 59^{4} +O(59^{5})\)
|
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.