Basic invariants
| Dimension: | $35$ |
| Group: | $S_7$ |
| Conductor: | \(932\!\cdots\!899\)\(\medspace = 396259^{15} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.1.396259.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 70 |
| Parity: | odd |
| Determinant: | 1.396259.2t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.1.396259.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - x^{4} - x^{3} + x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 103 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 103 }$:
\( x^{2} + 102x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 7 + 25\cdot 103 + 98\cdot 103^{2} + 36\cdot 103^{3} + 98\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 78 a + 69 + \left(66 a + 68\right)\cdot 103 + \left(48 a + 90\right)\cdot 103^{2} + \left(44 a + 93\right)\cdot 103^{3} + \left(16 a + 33\right)\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 4 a + 81 + \left(57 a + 5\right)\cdot 103 + \left(43 a + 70\right)\cdot 103^{2} + \left(74 a + 19\right)\cdot 103^{3} + \left(6 a + 80\right)\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 21 a + 1 + \left(30 a + 42\right)\cdot 103 + \left(96 a + 30\right)\cdot 103^{2} + 5\cdot 103^{3} + \left(53 a + 13\right)\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 25 a + 44 + \left(36 a + 57\right)\cdot 103 + \left(54 a + 72\right)\cdot 103^{2} + \left(58 a + 89\right)\cdot 103^{3} + \left(86 a + 5\right)\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 82 a + 22 + \left(72 a + 51\right)\cdot 103 + \left(6 a + 96\right)\cdot 103^{2} + \left(102 a + 12\right)\cdot 103^{3} + \left(49 a + 65\right)\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 99 a + 85 + \left(45 a + 58\right)\cdot 103 + \left(59 a + 56\right)\cdot 103^{2} + \left(28 a + 50\right)\cdot 103^{3} + \left(96 a + 12\right)\cdot 103^{4} +O(103^{5})\)
|
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 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.