Basic invariants
Dimension: | $21$ |
Group: | $S_7$ |
Conductor: | \(101\!\cdots\!375\)\(\medspace = 5^{11} \cdot 53^{11} \cdot 1327^{11} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.1.351655.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 42T418 |
Parity: | odd |
Determinant: | 1.351655.2t1.a.a |
Projective image: | $S_7$ |
Projective stem field: | Galois closure of 7.1.351655.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - x^{6} - 2x^{4} + x^{3} + x + 1 \) . |
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 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.