Basic invariants
| Dimension: | $20$ |
| Group: | $S_7$ |
| Conductor: | \(200\!\cdots\!009\)\(\medspace = 53^{12} \cdot 577^{10} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 7.3.1620793.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 70 |
| Parity: | even |
| Projective image: | $S_7$ |
| Projective field: | Galois closure of 7.3.1620793.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 199 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 199 }$:
\( x^{2} + 193x + 3 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 117 a + 15 + \left(57 a + 144\right)\cdot 199 + \left(38 a + 42\right)\cdot 199^{2} + \left(118 a + 191\right)\cdot 199^{3} + \left(106 a + 6\right)\cdot 199^{4} +O(199^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 136 a + 83 + \left(156 a + 60\right)\cdot 199 + \left(118 a + 95\right)\cdot 199^{2} + \left(119 a + 14\right)\cdot 199^{3} + \left(52 a + 162\right)\cdot 199^{4} +O(199^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 82 a + 120 + \left(141 a + 173\right)\cdot 199 + \left(160 a + 15\right)\cdot 199^{2} + \left(80 a + 66\right)\cdot 199^{3} + \left(92 a + 130\right)\cdot 199^{4} +O(199^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 112 a + 113 + \left(146 a + 179\right)\cdot 199 + \left(7 a + 169\right)\cdot 199^{2} + \left(34 a + 137\right)\cdot 199^{3} + \left(184 a + 93\right)\cdot 199^{4} +O(199^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 87 a + 188 + \left(52 a + 150\right)\cdot 199 + \left(191 a + 69\right)\cdot 199^{2} + \left(164 a + 135\right)\cdot 199^{3} + \left(14 a + 169\right)\cdot 199^{4} +O(199^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 177 + 18\cdot 199 + 149\cdot 199^{2} + 35\cdot 199^{3} + 74\cdot 199^{4} +O(199^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 63 a + 103 + \left(42 a + 68\right)\cdot 199 + \left(80 a + 54\right)\cdot 199^{2} + \left(79 a + 16\right)\cdot 199^{3} + \left(146 a + 159\right)\cdot 199^{4} +O(199^{5})\)
|
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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $20$ |
| $21$ | $2$ | $(1,2)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-4$ |
| $70$ | $3$ | $(1,2,3)$ | $2$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
| $210$ | $4$ | $(1,2,3,4)$ | $0$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $2$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $-1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $0$ |