Basic invariants
| Dimension: | $35$ |
| Group: | $S_7$ |
| Conductor: | \(147\!\cdots\!143\)\(\medspace = 29^{15} \cdot 43^{15} \cdot 241^{15} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.1.300527.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 70 |
| Parity: | odd |
| Determinant: | 1.300527.2t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.1.300527.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 2x^{6} + 2x^{5} - 2x^{3} + 4x^{2} - 3x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 193 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 193 }$:
\( x^{2} + 192x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 60 a + 4 + \left(135 a + 54\right)\cdot 193 + \left(147 a + 108\right)\cdot 193^{2} + \left(3 a + 104\right)\cdot 193^{3} + \left(a + 164\right)\cdot 193^{4} +O(193^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 133 a + 64 + \left(57 a + 129\right)\cdot 193 + \left(45 a + 120\right)\cdot 193^{2} + \left(189 a + 153\right)\cdot 193^{3} + \left(191 a + 161\right)\cdot 193^{4} +O(193^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 167 a + 190 + \left(35 a + 118\right)\cdot 193 + \left(24 a + 111\right)\cdot 193^{2} + \left(113 a + 131\right)\cdot 193^{3} + \left(87 a + 34\right)\cdot 193^{4} +O(193^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 78 + 154\cdot 193 + 18\cdot 193^{2} + 148\cdot 193^{3} + 158\cdot 193^{4} +O(193^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 26 a + 164 + \left(157 a + 180\right)\cdot 193 + \left(168 a + 99\right)\cdot 193^{2} + \left(79 a + 27\right)\cdot 193^{3} + \left(105 a + 9\right)\cdot 193^{4} +O(193^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 45 a + 18 + \left(175 a + 2\right)\cdot 193 + \left(46 a + 124\right)\cdot 193^{2} + \left(118 a + 67\right)\cdot 193^{3} + \left(169 a + 192\right)\cdot 193^{4} +O(193^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 148 a + 63 + \left(17 a + 132\right)\cdot 193 + \left(146 a + 188\right)\cdot 193^{2} + \left(74 a + 138\right)\cdot 193^{3} + \left(23 a + 50\right)\cdot 193^{4} +O(193^{5})\)
|
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.