Basic invariants
| Dimension: | $35$ |
| Group: | $S_7$ |
| Conductor: | \(101\!\cdots\!801\)\(\medspace = 109^{20} \cdot 269^{20} \cdot 2153^{20} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.7.63128113.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 126 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.7.63128113.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 2x^{6} - 5x^{5} + 8x^{4} + 6x^{3} - 7x^{2} - x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$:
\( x^{2} + 63x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 22 a + 51 + \left(46 a + 9\right)\cdot 67 + \left(35 a + 44\right)\cdot 67^{2} + 47 a\cdot 67^{3} + \left(42 a + 59\right)\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 28 + 67 + 61\cdot 67^{2} + 8\cdot 67^{3} + 13\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 23 a + 18 + \left(64 a + 11\right)\cdot 67 + \left(39 a + 3\right)\cdot 67^{2} + \left(44 a + 28\right)\cdot 67^{3} + \left(12 a + 50\right)\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 45 a + 5 + \left(20 a + 39\right)\cdot 67 + \left(31 a + 6\right)\cdot 67^{2} + \left(19 a + 21\right)\cdot 67^{3} + \left(24 a + 48\right)\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 50 a + 63 + \left(7 a + 56\right)\cdot 67 + \left(2 a + 26\right)\cdot 67^{2} + \left(55 a + 46\right)\cdot 67^{3} + \left(41 a + 64\right)\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 17 a + 62 + \left(59 a + 37\right)\cdot 67 + \left(64 a + 27\right)\cdot 67^{2} + \left(11 a + 63\right)\cdot 67^{3} + \left(25 a + 42\right)\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 44 a + 43 + \left(2 a + 44\right)\cdot 67 + \left(27 a + 31\right)\cdot 67^{2} + \left(22 a + 32\right)\cdot 67^{3} + \left(54 a + 56\right)\cdot 67^{4} +O(67^{5})\)
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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 |
| $1$ | $1$ | $()$ | $35$ |
| $21$ | $2$ | $(1,2)$ | $-5$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $70$ | $3$ | $(1,2,3)$ | $-1$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $210$ | $4$ | $(1,2,3,4)$ | $1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.