Basic invariants
| Dimension: | $6$ |
| Group: | $S_7$ |
| Conductor: | \(394\!\cdots\!207\)\(\medspace = 23^{5} \cdot 14369^{5} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.1.330487.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 14T46 |
| Parity: | odd |
| Determinant: | 1.330487.2t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.1.330487.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} + x^{5} - x^{3} - x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 521 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 521 }$:
\( x^{2} + 515x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 488 + 5\cdot 521 + 3\cdot 521^{2} + 208\cdot 521^{3} + 195\cdot 521^{4} +O(521^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 169 a + 518 + \left(14 a + 437\right)\cdot 521 + \left(326 a + 282\right)\cdot 521^{2} + \left(308 a + 342\right)\cdot 521^{3} + \left(289 a + 361\right)\cdot 521^{4} +O(521^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 155 + 361\cdot 521 + 468\cdot 521^{2} + 423\cdot 521^{3} + 373\cdot 521^{4} +O(521^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 352 a + 490 + \left(506 a + 354\right)\cdot 521 + \left(194 a + 140\right)\cdot 521^{2} + \left(212 a + 305\right)\cdot 521^{3} + \left(231 a + 227\right)\cdot 521^{4} +O(521^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 199 + 373\cdot 521 + 481\cdot 521^{2} + 3\cdot 521^{3} + 235\cdot 521^{4} +O(521^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 327 + 10\cdot 521 + 31\cdot 521^{2} + 419\cdot 521^{3} + 143\cdot 521^{4} +O(521^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 428 + 18\cdot 521 + 155\cdot 521^{2} + 381\cdot 521^{3} + 25\cdot 521^{4} +O(521^{5})\)
|
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.