Basic invariants
| Dimension: | $6$ |
| Group: | $S_7$ |
| Conductor: | \(107704601\)\(\medspace = 4283 \cdot 25147 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.7.107704601.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_7$ |
| Parity: | even |
| Determinant: | 1.107704601.2t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.7.107704601.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 2x^{6} - 6x^{5} + 12x^{4} + 4x^{3} - 8x^{2} - x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 691 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 691 }$:
\( x^{2} + 686x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 669 a + 538 + \left(527 a + 528\right)\cdot 691 + \left(206 a + 560\right)\cdot 691^{2} + \left(291 a + 544\right)\cdot 691^{3} + \left(211 a + 508\right)\cdot 691^{4} +O(691^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 148 + 57\cdot 691 + 302\cdot 691^{2} + 291\cdot 691^{3} + 521\cdot 691^{4} +O(691^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 354 a + 398 + \left(341 a + 203\right)\cdot 691 + \left(275 a + 601\right)\cdot 691^{2} + \left(597 a + 146\right)\cdot 691^{3} + \left(294 a + 324\right)\cdot 691^{4} +O(691^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 133 a + 247 + \left(146 a + 387\right)\cdot 691 + \left(326 a + 628\right)\cdot 691^{2} + \left(589 a + 362\right)\cdot 691^{3} + \left(220 a + 591\right)\cdot 691^{4} +O(691^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 22 a + 428 + \left(163 a + 426\right)\cdot 691 + \left(484 a + 375\right)\cdot 691^{2} + \left(399 a + 412\right)\cdot 691^{3} + \left(479 a + 583\right)\cdot 691^{4} +O(691^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 558 a + 221 + \left(544 a + 294\right)\cdot 691 + \left(364 a + 40\right)\cdot 691^{2} + \left(101 a + 220\right)\cdot 691^{3} + \left(470 a + 415\right)\cdot 691^{4} +O(691^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 337 a + 95 + \left(349 a + 175\right)\cdot 691 + \left(415 a + 255\right)\cdot 691^{2} + \left(93 a + 94\right)\cdot 691^{3} + \left(396 a + 510\right)\cdot 691^{4} +O(691^{5})\)
|
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.