Basic invariants
| Dimension: | $14$ |
| Group: | $S_7$ |
| Conductor: | \(217\!\cdots\!321\)\(\medspace = 701^{4} \cdot 1733^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.3.1214833.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 21T38 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.3.1214833.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - x^{6} - x^{5} + 2x^{4} + 3x^{3} - x^{2} - 3x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$:
\( x^{2} + 49x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 33 a + 30 + \left(44 a + 35\right)\cdot 53 + \left(25 a + 15\right)\cdot 53^{2} + 5 a\cdot 53^{3} + \left(36 a + 20\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 11 + 15\cdot 53 + 23\cdot 53^{2} + 18\cdot 53^{3} + 6\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 29 a + 11 + \left(10 a + 14\right)\cdot 53 + \left(37 a + 9\right)\cdot 53^{2} + 17\cdot 53^{3} + \left(18 a + 41\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 20 a + 3 + \left(8 a + 22\right)\cdot 53 + \left(27 a + 21\right)\cdot 53^{2} + \left(47 a + 49\right)\cdot 53^{3} + \left(16 a + 52\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 9 a + 24 + \left(20 a + 39\right)\cdot 53 + \left(9 a + 41\right)\cdot 53^{2} + \left(26 a + 50\right)\cdot 53^{3} + \left(35 a + 10\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 24 a + 21 + \left(42 a + 27\right)\cdot 53 + \left(15 a + 41\right)\cdot 53^{2} + \left(52 a + 35\right)\cdot 53^{3} + \left(34 a + 6\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 44 a + 7 + \left(32 a + 5\right)\cdot 53 + \left(43 a + 6\right)\cdot 53^{2} + \left(26 a + 40\right)\cdot 53^{3} + \left(17 a + 20\right)\cdot 53^{4} +O(53^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $14$ |
| $21$ | $2$ | $(1,2)$ | $6$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $2$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $2$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $210$ | $4$ | $(1,2,3,4)$ | $0$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $2$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.