Basic invariants
| Dimension: | $20$ |
| Group: | $S_7$ |
| Conductor: | \(132\!\cdots\!481\)\(\medspace = 11^{10} \cdot 13^{12} \cdot 859^{10} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.3.1596881.1 |
| 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 7.3.1596881.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 2x^{6} - 2x^{5} + 6x^{4} - x^{3} - 4x^{2} + 2x + 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$:
\( x^{2} + 97x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 15 a + 56 + \left(84 a + 50\right)\cdot 101 + \left(47 a + 98\right)\cdot 101^{2} + \left(99 a + 100\right)\cdot 101^{3} + \left(56 a + 87\right)\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 71 + 32\cdot 101 + 101^{2} + 34\cdot 101^{3} + 13\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 34 a + 85 + \left(14 a + 43\right)\cdot 101 + \left(98 a + 28\right)\cdot 101^{2} + \left(55 a + 97\right)\cdot 101^{3} + \left(36 a + 99\right)\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 26 + 5\cdot 101 + 88\cdot 101^{2} + 38\cdot 101^{3} + 9\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 67 a + 19 + \left(86 a + 67\right)\cdot 101 + \left(2 a + 2\right)\cdot 101^{2} + \left(45 a + 21\right)\cdot 101^{3} + \left(64 a + 89\right)\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 33 + 34\cdot 101 + 80\cdot 101^{2} + 64\cdot 101^{3} + 89\cdot 101^{4} +O(101^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 86 a + 15 + \left(16 a + 69\right)\cdot 101 + \left(53 a + 3\right)\cdot 101^{2} + \left(a + 47\right)\cdot 101^{3} + \left(44 a + 14\right)\cdot 101^{4} +O(101^{5})\)
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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 |
| $1$ | $1$ | $()$ | $20$ |
| $21$ | $2$ | $(1,2)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-4$ |
| $70$ | $3$ | $(1,2,3)$ | $2$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
| $210$ | $4$ | $(1,2,3,4)$ | $0$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $2$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $-1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.