Basic invariants
| Dimension: | $15$ |
| Group: | $S_7$ |
| Conductor: | \(537\!\cdots\!707\)\(\medspace = 351587^{5} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.1.351587.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 42T412 |
| Parity: | odd |
| Determinant: | 1.351587.2t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.1.351587.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 2x^{5} - 2x^{4} + 2x^{3} + 3x^{2} - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$:
\( x^{2} + 45x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 a + 9 + \left(34 a + 4\right)\cdot 47 + \left(6 a + 38\right)\cdot 47^{2} + \left(29 a + 29\right)\cdot 47^{3} + \left(24 a + 12\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 6 a + 32 + \left(32 a + 42\right)\cdot 47 + \left(20 a + 18\right)\cdot 47^{2} + \left(28 a + 33\right)\cdot 47^{3} + \left(20 a + 28\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 41 a + 44 + \left(14 a + 6\right)\cdot 47 + \left(26 a + 28\right)\cdot 47^{2} + \left(18 a + 22\right)\cdot 47^{3} + \left(26 a + 41\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 22 a + 8 + \left(46 a + 24\right)\cdot 47 + \left(32 a + 35\right)\cdot 47^{2} + \left(8 a + 3\right)\cdot 47^{3} + \left(40 a + 34\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 43 a + 17 + \left(12 a + 21\right)\cdot 47 + \left(40 a + 17\right)\cdot 47^{2} + \left(17 a + 34\right)\cdot 47^{3} + \left(22 a + 32\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 26 + 40\cdot 47 + 41\cdot 47^{2} + 28\cdot 47^{3} + 26\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 25 a + 5 + 47 + \left(14 a + 8\right)\cdot 47^{2} + \left(38 a + 35\right)\cdot 47^{3} + \left(6 a + 11\right)\cdot 47^{4} +O(47^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $15$ |
| $21$ | $2$ | $(1,2)$ | $5$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $70$ | $3$ | $(1,2,3)$ | $3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.