Basic invariants
| Dimension: | $20$ |
| Group: | $S_7$ |
| Conductor: | \(342\!\cdots\!456\)\(\medspace = 2^{48} \cdot 3^{40} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.1.90699264.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 70 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.1.90699264.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - x^{6} + 3x^{5} - 5x^{4} + 2x^{3} - 12x^{2} + x - 7 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 643 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 643 }$:
\( x^{2} + 641x + 11 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 531 a + 389 + \left(248 a + 398\right)\cdot 643 + \left(558 a + 593\right)\cdot 643^{2} + \left(630 a + 619\right)\cdot 643^{3} + \left(436 a + 272\right)\cdot 643^{4} +O(643^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 136 a + 132 + \left(330 a + 449\right)\cdot 643 + \left(384 a + 403\right)\cdot 643^{2} + \left(87 a + 478\right)\cdot 643^{3} + \left(302 a + 302\right)\cdot 643^{4} +O(643^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 507 a + 404 + \left(312 a + 330\right)\cdot 643 + \left(258 a + 199\right)\cdot 643^{2} + \left(555 a + 269\right)\cdot 643^{3} + \left(340 a + 176\right)\cdot 643^{4} +O(643^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 6 + 129\cdot 643 + 131\cdot 643^{2} + 554\cdot 643^{3} + 454\cdot 643^{4} +O(643^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 9 a + 408 + \left(620 a + 155\right)\cdot 643 + \left(602 a + 241\right)\cdot 643^{2} + \left(398 a + 530\right)\cdot 643^{3} + \left(17 a + 284\right)\cdot 643^{4} +O(643^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 112 a + 165 + \left(394 a + 365\right)\cdot 643 + \left(84 a + 175\right)\cdot 643^{2} + \left(12 a + 37\right)\cdot 643^{3} + \left(206 a + 516\right)\cdot 643^{4} +O(643^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 634 a + 426 + \left(22 a + 100\right)\cdot 643 + \left(40 a + 184\right)\cdot 643^{2} + \left(244 a + 82\right)\cdot 643^{3} + \left(625 a + 564\right)\cdot 643^{4} +O(643^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $20$ |
| $21$ | $2$ | $(1,2)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-4$ |
| $70$ | $3$ | $(1,2,3)$ | $2$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
| $210$ | $4$ | $(1,2,3,4)$ | $0$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $2$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $-1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.