Basic invariants
| Dimension: | $35$ |
| Group: | $S_7$ |
| Conductor: | \(867\!\cdots\!736\)\(\medspace = 2^{97} \cdot 3^{77} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.1.1451188224.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 70 |
| Parity: | odd |
| Determinant: | 1.24.2t1.b.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.1.1451188224.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 2x^{6} + 27x^{3} - 54x^{2} + 42x - 12 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 271 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 271 }$:
\( x^{2} + 269x + 6 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 125 a + 78 + \left(60 a + 145\right)\cdot 271 + \left(51 a + 155\right)\cdot 271^{2} + \left(22 a + 25\right)\cdot 271^{3} + \left(65 a + 192\right)\cdot 271^{4} +O(271^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 146 a + 57 + \left(210 a + 141\right)\cdot 271 + \left(219 a + 197\right)\cdot 271^{2} + \left(248 a + 18\right)\cdot 271^{3} + \left(205 a + 29\right)\cdot 271^{4} +O(271^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 162 a + 174 + \left(207 a + 74\right)\cdot 271 + \left(194 a + 137\right)\cdot 271^{2} + \left(265 a + 70\right)\cdot 271^{3} + \left(134 a + 216\right)\cdot 271^{4} +O(271^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 94 + 9\cdot 271 + 63\cdot 271^{2} + 11\cdot 271^{3} + 215\cdot 271^{4} +O(271^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 109 a + 227 + \left(63 a + 56\right)\cdot 271 + \left(76 a + 48\right)\cdot 271^{2} + \left(5 a + 136\right)\cdot 271^{3} + \left(136 a + 220\right)\cdot 271^{4} +O(271^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 131 + 72\cdot 271 + 119\cdot 271^{2} + 232\cdot 271^{3} + 26\cdot 271^{4} +O(271^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 54 + 42\cdot 271 + 92\cdot 271^{2} + 47\cdot 271^{3} + 184\cdot 271^{4} +O(271^{5})\)
|
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.