Basic invariants
| Dimension: | $6$ |
| Group: | $S_7$ |
| Conductor: | \(109\!\cdots\!857\)\(\medspace = 1018217^{5} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.3.1018217.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 14T46 |
| Parity: | even |
| Determinant: | 1.1018217.2t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.3.1018217.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - x^{6} + x^{5} - 3x^{4} - x^{2} + x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$:
\( x^{2} + 33x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 16 a + 13 + \left(27 a + 19\right)\cdot 37 + \left(18 a + 22\right)\cdot 37^{2} + \left(33 a + 8\right)\cdot 37^{3} + \left(2 a + 15\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 14 a + 12 + \left(29 a + 11\right)\cdot 37 + \left(7 a + 28\right)\cdot 37^{2} + \left(5 a + 34\right)\cdot 37^{3} + \left(15 a + 16\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 21 a + 3 + \left(9 a + 2\right)\cdot 37 + \left(18 a + 33\right)\cdot 37^{2} + \left(3 a + 12\right)\cdot 37^{3} + \left(34 a + 30\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 18 a + 32 + \left(2 a + 32\right)\cdot 37 + \left(22 a + 9\right)\cdot 37^{2} + \left(6 a + 36\right)\cdot 37^{3} + \left(8 a + 27\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 23 a + 31 + \left(7 a + 3\right)\cdot 37 + \left(29 a + 30\right)\cdot 37^{2} + \left(31 a + 10\right)\cdot 37^{3} + \left(21 a + 35\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 19 a + 30 + \left(34 a + 24\right)\cdot 37 + \left(14 a + 21\right)\cdot 37^{2} + \left(30 a + 3\right)\cdot 37^{3} + \left(28 a + 17\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 28 + 16\cdot 37 + 2\cdot 37^{2} + 4\cdot 37^{3} + 5\cdot 37^{4} +O(37^{5})\)
|
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 | Complex conjugation |
| $1$ | $1$ | $()$ | $6$ | |
| $21$ | $2$ | $(1,2)$ | $-4$ | |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ | |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ | ✓ |
| $70$ | $3$ | $(1,2,3)$ | $3$ | |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ | |
| $210$ | $4$ | $(1,2,3,4)$ | $-2$ | |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ | |
| $504$ | $5$ | $(1,2,3,4,5)$ | $1$ | |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ | |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ | |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ | |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $-1$ | |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $1$ | |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $1$ |