Basic invariants
| Dimension: | $6$ |
| Group: | $S_7$ |
| Conductor: | \(77004029\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.7.77004029.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_7$ |
| Parity: | even |
| Determinant: | 1.77004029.2t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.7.77004029.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - x^{6} - 7x^{5} + 4x^{4} + 15x^{3} - 2x^{2} - 9x - 2 \)
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The roots of $f$ are computed in an extension of $\Q_{ 89 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 89 }$:
\( x^{2} + 82x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 52 a + 77 + \left(9 a + 33\right)\cdot 89 + \left(28 a + 66\right)\cdot 89^{2} + \left(61 a + 88\right)\cdot 89^{3} + \left(51 a + 37\right)\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 19 + 12\cdot 89 + 13\cdot 89^{2} + 63\cdot 89^{3} + 8\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 37 a + 85 + \left(79 a + 48\right)\cdot 89 + \left(60 a + 75\right)\cdot 89^{2} + \left(27 a + 44\right)\cdot 89^{3} + \left(37 a + 71\right)\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 49 a + 41 + \left(19 a + 73\right)\cdot 89 + 20 a\cdot 89^{2} + \left(10 a + 81\right)\cdot 89^{3} + \left(75 a + 66\right)\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 40 a + 28 + \left(69 a + 72\right)\cdot 89 + \left(68 a + 33\right)\cdot 89^{2} + \left(78 a + 43\right)\cdot 89^{3} + \left(13 a + 48\right)\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 49 + 46\cdot 89 + 3\cdot 89^{2} + 72\cdot 89^{3} + 28\cdot 89^{4} +O(89^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 58 + 68\cdot 89 + 73\cdot 89^{2} + 51\cdot 89^{3} + 4\cdot 89^{4} +O(89^{5})\)
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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.