Basic invariants
| Dimension: | $20$ |
| Group: | $S_7$ |
| Conductor: | \(729\!\cdots\!649\)\(\medspace = 193327^{10} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.1.193327.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 70 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.1.193327.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - x^{6} + 2x^{4} - 2x^{3} + 2x - 1 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 2 + 22\cdot 83 + 22\cdot 83^{2} + 70\cdot 83^{3} + 8\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 32 + 54\cdot 83 + 66\cdot 83^{2} + 19\cdot 83^{3} + 75\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 66 + 10\cdot 83^{2} + 59\cdot 83^{3} + 7\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 16 a + 37 + \left(78 a + 41\right)\cdot 83 + \left(47 a + 58\right)\cdot 83^{2} + \left(70 a + 28\right)\cdot 83^{3} + \left(58 a + 81\right)\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 82 a + 72 + \left(43 a + 73\right)\cdot 83 + \left(14 a + 4\right)\cdot 83^{2} + \left(25 a + 46\right)\cdot 83^{3} + \left(19 a + 47\right)\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( a + 71 + \left(39 a + 35\right)\cdot 83 + \left(68 a + 58\right)\cdot 83^{2} + \left(57 a + 56\right)\cdot 83^{3} + \left(63 a + 41\right)\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 67 a + 53 + \left(4 a + 20\right)\cdot 83 + \left(35 a + 28\right)\cdot 83^{2} + \left(12 a + 51\right)\cdot 83^{3} + \left(24 a + 69\right)\cdot 83^{4} +O(83^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $20$ |
| $21$ | $2$ | $(1,2)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-4$ |
| $70$ | $3$ | $(1,2,3)$ | $2$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
| $210$ | $4$ | $(1,2,3,4)$ | $0$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $2$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $-1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.