Basic invariants
| Dimension: | $20$ |
| Group: | $S_7$ |
| Conductor: | \(340\!\cdots\!049\)\(\medspace = 7^{12} \cdot 31^{10} \cdot 353^{10} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.1.536207.1 |
| 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 7.1.536207.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - x^{6} + 2x^{5} + x^{4} + 2x^{2} + x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 191 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 191 }$:
\( x^{2} + 190x + 19 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 60 a + 1 + \left(105 a + 139\right)\cdot 191 + \left(176 a + 128\right)\cdot 191^{2} + \left(42 a + 3\right)\cdot 191^{3} + \left(20 a + 178\right)\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 108 a + 37 + \left(23 a + 160\right)\cdot 191 + \left(40 a + 82\right)\cdot 191^{2} + \left(179 a + 169\right)\cdot 191^{3} + \left(76 a + 182\right)\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 45 + 72\cdot 191 + 169\cdot 191^{2} + 35\cdot 191^{3} + 33\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 83 a + 145 + \left(167 a + 75\right)\cdot 191 + \left(150 a + 99\right)\cdot 191^{2} + \left(11 a + 117\right)\cdot 191^{3} + \left(114 a + 80\right)\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 10 + 50\cdot 191 + 99\cdot 191^{2} + 181\cdot 191^{3} + 41\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 131 a + 61 + \left(85 a + 184\right)\cdot 191 + \left(14 a + 8\right)\cdot 191^{2} + \left(148 a + 61\right)\cdot 191^{3} + \left(170 a + 155\right)\cdot 191^{4} +O(191^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 84 + 82\cdot 191 + 175\cdot 191^{2} + 3\cdot 191^{3} + 92\cdot 191^{4} +O(191^{5})\)
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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.