Basic invariants
| Dimension: | $6$ |
| Group: | $S_7$ |
| Conductor: | \(283223\)\(\medspace = 61 \cdot 4643 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.1.283223.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_7$ |
| Parity: | odd |
| Determinant: | 1.283223.2t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.1.283223.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} + x^{3} - 2x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 251 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 251 }$:
\( x^{2} + 242x + 6 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 151 a + 29 + \left(204 a + 186\right)\cdot 251 + \left(33 a + 43\right)\cdot 251^{2} + \left(43 a + 66\right)\cdot 251^{3} + \left(127 a + 169\right)\cdot 251^{4} +O(251^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 239 + 165\cdot 251 + 223\cdot 251^{2} + 181\cdot 251^{3} + 187\cdot 251^{4} +O(251^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 247 a + 57 + \left(15 a + 75\right)\cdot 251 + \left(104 a + 24\right)\cdot 251^{2} + \left(81 a + 75\right)\cdot 251^{3} + \left(110 a + 186\right)\cdot 251^{4} +O(251^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 132 + 145\cdot 251 + 242\cdot 251^{2} + 242\cdot 251^{3} + 26\cdot 251^{4} +O(251^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 100 a + 133 + \left(46 a + 119\right)\cdot 251 + \left(217 a + 143\right)\cdot 251^{2} + \left(207 a + 169\right)\cdot 251^{3} + \left(123 a + 15\right)\cdot 251^{4} +O(251^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 4 a + 21 + \left(235 a + 223\right)\cdot 251 + \left(146 a + 191\right)\cdot 251^{2} + \left(169 a + 201\right)\cdot 251^{3} + \left(140 a + 93\right)\cdot 251^{4} +O(251^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 142 + 88\cdot 251 + 134\cdot 251^{2} + 66\cdot 251^{3} + 73\cdot 251^{4} +O(251^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $6$ |
| $21$ | $2$ | $(1,2)$ | $4$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $2$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $-1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $-1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.