Basic invariants
| Dimension: | $21$ |
| Group: | $S_7$ |
| Conductor: | \(159\!\cdots\!136\)\(\medspace = 2^{54} \cdot 3^{46} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.1.241864704.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 42T418 |
| Parity: | odd |
| Determinant: | 1.4.2t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.1.241864704.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 2x^{6} + 3x^{5} - 4x^{4} + 5x^{3} - 6x^{2} + 7x - 8 \)
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The roots of $f$ are computed in an extension of $\Q_{ 557 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 557 }$:
\( x^{2} + 553x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 62 a + 548 + \left(460 a + 127\right)\cdot 557 + \left(309 a + 40\right)\cdot 557^{2} + \left(318 a + 472\right)\cdot 557^{3} + \left(484 a + 347\right)\cdot 557^{4} +O(557^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 542 a + 168 + \left(436 a + 109\right)\cdot 557 + \left(503 a + 179\right)\cdot 557^{2} + \left(548 a + 180\right)\cdot 557^{3} + \left(523 a + 350\right)\cdot 557^{4} +O(557^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 420 + 47\cdot 557 + 197\cdot 557^{2} + 416\cdot 557^{3} + 260\cdot 557^{4} +O(557^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 495 a + 239 + \left(96 a + 235\right)\cdot 557 + \left(247 a + 262\right)\cdot 557^{2} + \left(238 a + 322\right)\cdot 557^{3} + \left(72 a + 296\right)\cdot 557^{4} +O(557^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 183 + 406\cdot 557 + 392\cdot 557^{2} + 5\cdot 557^{3} + 451\cdot 557^{4} +O(557^{5})\)
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| $r_{ 6 }$ | $=$ |
\( 7 + 543\cdot 557 + 512\cdot 557^{2} + 72\cdot 557^{3} + 295\cdot 557^{4} +O(557^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 15 a + 108 + \left(120 a + 201\right)\cdot 557 + \left(53 a + 86\right)\cdot 557^{2} + \left(8 a + 201\right)\cdot 557^{3} + \left(33 a + 226\right)\cdot 557^{4} +O(557^{5})\)
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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.