Basic invariants
| Dimension: | $21$ |
| Group: | $S_7$ |
| Conductor: | \(753\!\cdots\!169\)\(\medspace = 3^{11} \cdot 251^{11} \cdot 69073^{11} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.7.52011969.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 42T418 |
| Parity: | even |
| Determinant: | 1.52011969.2t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.7.52011969.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 3x^{6} - 3x^{5} + 13x^{4} - x^{3} - 13x^{2} + 4x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 239 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 239 }$:
\( x^{2} + 237x + 7 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 7 a + 6 + \left(145 a + 28\right)\cdot 239 + \left(209 a + 121\right)\cdot 239^{2} + \left(204 a + 32\right)\cdot 239^{3} + \left(233 a + 77\right)\cdot 239^{4} +O(239^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 217 a + 199 + \left(92 a + 213\right)\cdot 239 + \left(144 a + 26\right)\cdot 239^{2} + \left(64 a + 112\right)\cdot 239^{3} + \left(60 a + 149\right)\cdot 239^{4} +O(239^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 232 a + 20 + \left(93 a + 72\right)\cdot 239 + \left(29 a + 156\right)\cdot 239^{2} + \left(34 a + 232\right)\cdot 239^{3} + \left(5 a + 100\right)\cdot 239^{4} +O(239^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 134 + 93\cdot 239 + 122\cdot 239^{2} + 117\cdot 239^{3} + 200\cdot 239^{4} +O(239^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 22 a + 155 + \left(146 a + 182\right)\cdot 239 + \left(94 a + 222\right)\cdot 239^{2} + \left(174 a + 96\right)\cdot 239^{3} + \left(178 a + 205\right)\cdot 239^{4} +O(239^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 27 + 235\cdot 239 + 68\cdot 239^{2} + 41\cdot 239^{3} + 195\cdot 239^{4} +O(239^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 179 + 130\cdot 239 + 237\cdot 239^{2} + 83\cdot 239^{3} + 27\cdot 239^{4} +O(239^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $21$ |
| $21$ | $2$ | $(1,2)$ | $-1$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $3$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $70$ | $3$ | $(1,2,3)$ | $-3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $-1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.