Basic invariants
| Dimension: | $6$ |
| Group: | $S_7$ |
| Conductor: | \(231\!\cdots\!159\)\(\medspace = 3^{5} \cdot 53^{2} \cdot 1277^{5} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.5.10761279.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 14T46 |
| Parity: | odd |
| Determinant: | 1.3831.2t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.5.10761279.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - x^{6} - 2x^{5} + 4x^{4} - 3x^{3} - 4x^{2} + 5x - 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 211 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 211 }$:
\( x^{2} + 207x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 97 a + 191 + \left(80 a + 82\right)\cdot 211 + \left(130 a + 210\right)\cdot 211^{2} + \left(53 a + 158\right)\cdot 211^{3} + \left(31 a + 102\right)\cdot 211^{4} +O(211^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 187 a + 129 + \left(38 a + 106\right)\cdot 211 + \left(94 a + 96\right)\cdot 211^{2} + \left(93 a + 128\right)\cdot 211^{3} + \left(2 a + 111\right)\cdot 211^{4} +O(211^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 177 + 170\cdot 211 + 195\cdot 211^{2} + 184\cdot 211^{3} + 46\cdot 211^{4} +O(211^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 24 a + 33 + \left(172 a + 75\right)\cdot 211 + \left(116 a + 12\right)\cdot 211^{2} + \left(117 a + 197\right)\cdot 211^{3} + \left(208 a + 27\right)\cdot 211^{4} +O(211^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 38 + 11\cdot 211 + 43\cdot 211^{2} + 115\cdot 211^{3} + 32\cdot 211^{4} +O(211^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 114 a + 157 + \left(130 a + 96\right)\cdot 211 + \left(80 a + 18\right)\cdot 211^{2} + \left(157 a + 32\right)\cdot 211^{3} + \left(179 a + 174\right)\cdot 211^{4} +O(211^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 120 + 89\cdot 211 + 56\cdot 211^{2} + 27\cdot 211^{3} + 137\cdot 211^{4} +O(211^{5})\)
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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 |
| $1$ | $1$ | $()$ | $6$ |
| $21$ | $2$ | $(1,2)$ | $-4$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $-2$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $-1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.