Basic invariants
| Dimension: | $21$ |
| Group: | $S_7$ |
| Conductor: | \(739\!\cdots\!249\)\(\medspace = 193607^{10} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.1.193607.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 84 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.1.193607.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - x^{4} - x^{3} + x^{2} + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$:
\( x^{2} + 69x + 7 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 69 a + 69 + \left(67 a + 48\right)\cdot 71 + \left(13 a + 41\right)\cdot 71^{2} + \left(65 a + 56\right)\cdot 71^{3} + \left(32 a + 1\right)\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 54 + 34\cdot 71 + 41\cdot 71^{2} + 48\cdot 71^{3} + 15\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 68 a + 66 + \left(23 a + 36\right)\cdot 71 + \left(14 a + 45\right)\cdot 71^{2} + \left(22 a + 69\right)\cdot 71^{3} + \left(17 a + 21\right)\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 2 a + 65 + \left(3 a + 44\right)\cdot 71 + \left(57 a + 1\right)\cdot 71^{2} + \left(5 a + 31\right)\cdot 71^{3} + \left(38 a + 2\right)\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 12 + 15\cdot 71 + 56\cdot 71^{2} + 27\cdot 71^{3} + 10\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 3 a + 60 + \left(47 a + 16\right)\cdot 71 + \left(56 a + 50\right)\cdot 71^{2} + \left(48 a + 28\right)\cdot 71^{3} + \left(53 a + 34\right)\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 29 + 15\cdot 71 + 47\cdot 71^{2} + 21\cdot 71^{3} + 55\cdot 71^{4} +O(71^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $21$ |
| $21$ | $2$ | $(1,2)$ | $1$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $70$ | $3$ | $(1,2,3)$ | $-3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.