Basic invariants
| Dimension: | $35$ |
| Group: | $S_7$ |
| Conductor: | \(586\!\cdots\!423\)\(\medspace = 3^{48} \cdot 7207^{15} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.1.583767.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 70 |
| Parity: | odd |
| Determinant: | 1.7207.2t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.1.583767.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - x^{6} - x^{4} + x + 1 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 99 a + 97 + \left(13 a + 55\right)\cdot 103 + \left(96 a + 71\right)\cdot 103^{2} + \left(61 a + 5\right)\cdot 103^{3} + \left(33 a + 30\right)\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 22 + 38\cdot 103 + 40\cdot 103^{2} + 82\cdot 103^{3} + 72\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 25 a + 73 + \left(74 a + 74\right)\cdot 103 + \left(90 a + 41\right)\cdot 103^{2} + \left(58 a + 1\right)\cdot 103^{3} + \left(10 a + 97\right)\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 24 a + 3 + \left(9 a + 30\right)\cdot 103 + \left(10 a + 74\right)\cdot 103^{2} + \left(45 a + 18\right)\cdot 103^{3} + \left(26 a + 90\right)\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 79 a + 27 + \left(93 a + 15\right)\cdot 103 + \left(92 a + 75\right)\cdot 103^{2} + \left(57 a + 53\right)\cdot 103^{3} + \left(76 a + 71\right)\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 78 a + 98 + \left(28 a + 20\right)\cdot 103 + \left(12 a + 58\right)\cdot 103^{2} + \left(44 a + 72\right)\cdot 103^{3} + \left(92 a + 48\right)\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 4 a + 93 + \left(89 a + 73\right)\cdot 103 + \left(6 a + 50\right)\cdot 103^{2} + \left(41 a + 74\right)\cdot 103^{3} + \left(69 a + 1\right)\cdot 103^{4} +O(103^{5})\)
|
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.