Basic invariants
| Dimension: | $6$ |
| Group: | $S_7$ |
| Conductor: | \(247\!\cdots\!601\)\(\medspace = 7^{5} \cdot 263^{5} \cdot 41081^{5} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.7.75630121.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 14T46 |
| Parity: | even |
| Determinant: | 1.75630121.2t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.7.75630121.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - x^{6} - 8x^{5} + 9x^{4} + 16x^{3} - 18x^{2} - 7x + 7 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 64 a + 64 + \left(41 a + 63\right)\cdot 71 + \left(34 a + 12\right)\cdot 71^{2} + \left(69 a + 51\right)\cdot 71^{3} + \left(67 a + 20\right)\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 40 a + 22 + \left(a + 63\right)\cdot 71 + \left(45 a + 52\right)\cdot 71^{2} + \left(68 a + 44\right)\cdot 71^{3} + \left(31 a + 29\right)\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 7 a + 50 + \left(29 a + 12\right)\cdot 71 + \left(36 a + 40\right)\cdot 71^{2} + \left(a + 13\right)\cdot 71^{3} + \left(3 a + 16\right)\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 26 a + \left(25 a + 11\right)\cdot 71 + \left(58 a + 35\right)\cdot 71^{2} + \left(19 a + 68\right)\cdot 71^{3} + 61 a\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 45 a + 52 + \left(45 a + 35\right)\cdot 71 + \left(12 a + 55\right)\cdot 71^{2} + \left(51 a + 49\right)\cdot 71^{3} + \left(9 a + 32\right)\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 31 a + 31 + \left(69 a + 26\right)\cdot 71 + \left(25 a + 70\right)\cdot 71^{2} + \left(2 a + 65\right)\cdot 71^{3} + \left(39 a + 24\right)\cdot 71^{4} +O(71^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 66 + 70\cdot 71 + 16\cdot 71^{2} + 61\cdot 71^{3} + 16\cdot 71^{4} +O(71^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $6$ |
| $21$ | $2$ | $(1,2)$ | $-4$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $-2$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $-1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.