Basic invariants
| Dimension: | $6$ |
| Group: | $S_7$ |
| Conductor: | \(315631\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.1.315631.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_7$ |
| Parity: | odd |
| Determinant: | 1.315631.2t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.1.315631.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - x^{5} - 2x^{4} + x^{2} + x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 163 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 163 }$:
\( x^{2} + 159x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 46 + 120\cdot 163 + 63\cdot 163^{2} + 62\cdot 163^{3} + 5\cdot 163^{4} +O(163^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 100 a + 25 + \left(71 a + 81\right)\cdot 163 + \left(39 a + 131\right)\cdot 163^{2} + \left(2 a + 129\right)\cdot 163^{3} + \left(66 a + 60\right)\cdot 163^{4} +O(163^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 38 a + 23 + \left(162 a + 69\right)\cdot 163 + \left(76 a + 150\right)\cdot 163^{2} + \left(49 a + 65\right)\cdot 163^{3} + \left(132 a + 143\right)\cdot 163^{4} +O(163^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 109 + 86\cdot 163 + 9\cdot 163^{2} + 100\cdot 163^{3} + 126\cdot 163^{4} +O(163^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 12 + 162\cdot 163 + 108\cdot 163^{2} + 7\cdot 163^{3} + 22\cdot 163^{4} +O(163^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 125 a + 12 + 28\cdot 163 + \left(86 a + 133\right)\cdot 163^{2} + \left(113 a + 23\right)\cdot 163^{3} + \left(30 a + 134\right)\cdot 163^{4} +O(163^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 63 a + 99 + \left(91 a + 104\right)\cdot 163 + \left(123 a + 54\right)\cdot 163^{2} + \left(160 a + 99\right)\cdot 163^{3} + \left(96 a + 159\right)\cdot 163^{4} +O(163^{5})\)
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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.