Basic invariants
| Dimension: | $35$ |
| Group: | $S_7$ |
| Conductor: | \(495\!\cdots\!623\)\(\medspace = 3^{48} \cdot 7127^{15} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.1.577287.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 70 |
| Parity: | odd |
| Determinant: | 1.7127.2t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.1.577287.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 2x^{6} + x^{5} + x^{2} - x - 1 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 39 a + 28 + \left(16 a + 38\right)\cdot 47 + \left(5 a + 35\right)\cdot 47^{2} + \left(30 a + 20\right)\cdot 47^{3} + \left(45 a + 10\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 42 a + 26 + \left(35 a + 5\right)\cdot 47 + \left(9 a + 9\right)\cdot 47^{2} + \left(31 a + 44\right)\cdot 47^{3} + \left(4 a + 40\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 27 + 18\cdot 47 + 44\cdot 47^{2} + 46\cdot 47^{3} + 8\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 5 a + 16 + \left(11 a + 35\right)\cdot 47 + \left(37 a + 39\right)\cdot 47^{2} + \left(15 a + 2\right)\cdot 47^{3} + \left(42 a + 19\right)\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 42 + 30\cdot 47 + 46\cdot 47^{2} + 38\cdot 47^{3} + 45\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 39 + 25\cdot 47 + 29\cdot 47^{2} + 5\cdot 47^{3} + 38\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 8 a + 12 + \left(30 a + 33\right)\cdot 47 + \left(41 a + 29\right)\cdot 47^{2} + \left(16 a + 28\right)\cdot 47^{3} + \left(a + 24\right)\cdot 47^{4} +O(47^{5})\)
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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 |
| $1$ | $1$ | $()$ | $35$ |
| $21$ | $2$ | $(1,2)$ | $5$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $1$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $70$ | $3$ | $(1,2,3)$ | $-1$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $210$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $1$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.