Basic invariants
| Dimension: | $14$ |
| Group: | $S_7$ |
| Conductor: | \(523\!\cdots\!673\)\(\medspace = 3^{6} \cdot 43^{9} \cdot 173539^{9} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.7.67159593.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 30T565 |
| Parity: | even |
| Determinant: | 1.7462177.2t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.7.67159593.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 3x^{6} - 3x^{5} + 11x^{4} + 2x^{3} - 10x^{2} + 1 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 12 a + 55 + \left(80 a + 50\right)\cdot 83 + \left(8 a + 1\right)\cdot 83^{2} + \left(21 a + 53\right)\cdot 83^{3} + \left(17 a + 13\right)\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 75 a + 22 + \left(17 a + 34\right)\cdot 83 + \left(81 a + 32\right)\cdot 83^{2} + \left(15 a + 43\right)\cdot 83^{3} + \left(55 a + 52\right)\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 23 a + 58 + \left(15 a + 45\right)\cdot 83 + \left(66 a + 53\right)\cdot 83^{2} + \left(33 a + 76\right)\cdot 83^{3} + \left(5 a + 33\right)\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 38 + 67\cdot 83 + 30\cdot 83^{2} + 71\cdot 83^{3} + 41\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 60 a + 81 + \left(67 a + 37\right)\cdot 83 + \left(16 a + 21\right)\cdot 83^{2} + \left(49 a + 44\right)\cdot 83^{3} + \left(77 a + 5\right)\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 71 a + 67 + \left(2 a + 35\right)\cdot 83 + \left(74 a + 13\right)\cdot 83^{2} + \left(61 a + 65\right)\cdot 83^{3} + \left(65 a + 9\right)\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 8 a + 14 + \left(65 a + 60\right)\cdot 83 + \left(a + 12\right)\cdot 83^{2} + \left(67 a + 61\right)\cdot 83^{3} + \left(27 a + 8\right)\cdot 83^{4} +O(83^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $14$ |
| $21$ | $2$ | $(1,2)$ | $-4$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $-1$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
| $210$ | $4$ | $(1,2,3,4)$ | $2$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.