Basic invariants
| Dimension: | $6$ |
| Group: | $S_7$ |
| Conductor: | \(409091\)\(\medspace = 313 \cdot 1307 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.1.409091.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_7$ |
| Parity: | odd |
| Determinant: | 1.409091.2t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.1.409091.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - x^{5} + 2x^{3} - 2x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 353 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 353 }$:
\( x^{2} + 348x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 232 a + 227 + \left(27 a + 2\right)\cdot 353 + \left(298 a + 81\right)\cdot 353^{2} + \left(349 a + 121\right)\cdot 353^{3} + \left(152 a + 338\right)\cdot 353^{4} +O(353^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 118 a + 151 + \left(107 a + 162\right)\cdot 353 + \left(302 a + 132\right)\cdot 353^{2} + \left(134 a + 105\right)\cdot 353^{3} + \left(84 a + 279\right)\cdot 353^{4} +O(353^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 243 + 159\cdot 353 + 14\cdot 353^{2} + 247\cdot 353^{3} + 192\cdot 353^{4} +O(353^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 121 a + 328 + \left(325 a + 261\right)\cdot 353 + \left(54 a + 131\right)\cdot 353^{2} + \left(3 a + 160\right)\cdot 353^{3} + \left(200 a + 47\right)\cdot 353^{4} +O(353^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 130 + 229\cdot 353 + 177\cdot 353^{2} + 315\cdot 353^{3} + 193\cdot 353^{4} +O(353^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 298 + 14\cdot 353 + 44\cdot 353^{2} + 338\cdot 353^{3} + 146\cdot 353^{4} +O(353^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 235 a + 35 + \left(245 a + 228\right)\cdot 353 + \left(50 a + 124\right)\cdot 353^{2} + \left(218 a + 124\right)\cdot 353^{3} + \left(268 a + 213\right)\cdot 353^{4} +O(353^{5})\)
|
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.