Basic invariants
| Dimension: | $21$ |
| Group: | $S_7$ |
| Conductor: | \(282\!\cdots\!649\)\(\medspace = 5560327^{10} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 7.5.5560327.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 84 |
| Parity: | even |
| Projective image: | $S_7$ |
| Projective field: | Galois closure of 7.5.5560327.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 293 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 293 }$:
\( x^{2} + 292x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 208 a + 94 + \left(264 a + 111\right)\cdot 293 + \left(139 a + 118\right)\cdot 293^{2} + \left(22 a + 221\right)\cdot 293^{3} + \left(192 a + 127\right)\cdot 293^{4} +O(293^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 85 a + 9 + \left(28 a + 168\right)\cdot 293 + \left(153 a + 286\right)\cdot 293^{2} + \left(270 a + 103\right)\cdot 293^{3} + \left(100 a + 4\right)\cdot 293^{4} +O(293^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 115 + 50\cdot 293 + 27\cdot 293^{2} + 166\cdot 293^{3} + 284\cdot 293^{4} +O(293^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 137 a + 152 + \left(282 a + 250\right)\cdot 293 + \left(107 a + 10\right)\cdot 293^{2} + \left(235 a + 185\right)\cdot 293^{3} + \left(88 a + 85\right)\cdot 293^{4} +O(293^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 182 a + 20 + \left(145 a + 116\right)\cdot 293 + \left(145 a + 153\right)\cdot 293^{2} + \left(78 a + 271\right)\cdot 293^{3} + \left(167 a + 27\right)\cdot 293^{4} +O(293^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 111 a + 202 + \left(147 a + 79\right)\cdot 293 + \left(147 a + 153\right)\cdot 293^{2} + \left(214 a + 204\right)\cdot 293^{3} + \left(125 a + 116\right)\cdot 293^{4} +O(293^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 156 a + 289 + \left(10 a + 102\right)\cdot 293 + \left(185 a + 129\right)\cdot 293^{2} + \left(57 a + 19\right)\cdot 293^{3} + \left(204 a + 232\right)\cdot 293^{4} +O(293^{5})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $21$ |
| $21$ | $2$ | $(1,2)$ | $1$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $70$ | $3$ | $(1,2,3)$ | $-3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |