Basic invariants
| Dimension: | $6$ |
| Group: | $S_7$ |
| Conductor: | \(351655\)\(\medspace = 5 \cdot 53 \cdot 1327 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 7.1.351655.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_7$ |
| Parity: | odd |
| Projective image: | $S_7$ |
| Projective field: | Galois closure of 7.1.351655.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 659 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 198 + 23\cdot 659 + 152\cdot 659^{2} + 170\cdot 659^{3} + 92\cdot 659^{4} +O(659^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 307 + 109\cdot 659 + 175\cdot 659^{2} + 260\cdot 659^{3} + 652\cdot 659^{4} +O(659^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 312 + 7\cdot 659 + 100\cdot 659^{2} + 302\cdot 659^{3} + 90\cdot 659^{4} +O(659^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 357 + 397\cdot 659 + 594\cdot 659^{2} + 483\cdot 659^{3} + 535\cdot 659^{4} +O(659^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 457 + 9\cdot 659 + 2\cdot 659^{2} + 251\cdot 659^{3} + 433\cdot 659^{4} +O(659^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 480 + 621\cdot 659 + 17\cdot 659^{2} + 489\cdot 659^{3} + 569\cdot 659^{4} +O(659^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 526 + 148\cdot 659 + 276\cdot 659^{2} + 20\cdot 659^{3} + 262\cdot 659^{4} +O(659^{5})\)
|
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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $6$ |
| $21$ | $2$ | $(1,2)$ | $4$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $2$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $-1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $-1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |