Basic invariants
| Dimension: | $35$ |
| Group: | $S_7$ |
| Conductor: | \(896\!\cdots\!817\)\(\medspace = 11^{18} \cdot 14033^{15} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.3.1697993.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 70 |
| Parity: | even |
| Determinant: | 1.14033.2t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.3.1697993.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 2x^{5} - 4x^{4} - x^{3} + 4x^{2} + 4x + 1 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 39 a + 167 + \left(92 a + 133\right)\cdot 251 + \left(227 a + 153\right)\cdot 251^{2} + \left(118 a + 139\right)\cdot 251^{3} + \left(65 a + 25\right)\cdot 251^{4} +O(251^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 159 a + 12 + \left(105 a + 79\right)\cdot 251 + \left(39 a + 74\right)\cdot 251^{2} + \left(180 a + 184\right)\cdot 251^{3} + \left(203 a + 25\right)\cdot 251^{4} +O(251^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 134 a + 32 + \left(79 a + 14\right)\cdot 251 + \left(182 a + 62\right)\cdot 251^{2} + \left(139 a + 105\right)\cdot 251^{3} + \left(248 a + 71\right)\cdot 251^{4} +O(251^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 92 a + 188 + \left(145 a + 117\right)\cdot 251 + \left(211 a + 72\right)\cdot 251^{2} + \left(70 a + 9\right)\cdot 251^{3} + \left(47 a + 173\right)\cdot 251^{4} +O(251^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 117 a + 234 + \left(171 a + 93\right)\cdot 251 + \left(68 a + 117\right)\cdot 251^{2} + \left(111 a + 176\right)\cdot 251^{3} + \left(2 a + 160\right)\cdot 251^{4} +O(251^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 104 + 143\cdot 251 + 173\cdot 251^{2} + 159\cdot 251^{3} + 51\cdot 251^{4} +O(251^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 212 a + 16 + \left(158 a + 171\right)\cdot 251 + \left(23 a + 99\right)\cdot 251^{2} + \left(132 a + 229\right)\cdot 251^{3} + \left(185 a + 244\right)\cdot 251^{4} +O(251^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $35$ |
| $21$ | $2$ | $(1,2)$ | $5$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $1$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $70$ | $3$ | $(1,2,3)$ | $-1$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $210$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $1$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.