Basic invariants
| Dimension: | $21$ |
| Group: | $S_7$ |
| Conductor: | \(275\!\cdots\!401\)\(\medspace = 11^{10} \cdot 577^{10} \cdot 11003^{10} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.7.69836041.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 84 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.7.69836041.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 3x^{6} - 4x^{5} + 16x^{4} - 20x^{2} + 8x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 a + 28 + \left(10 a + 30\right)\cdot 31 + \left(17 a + 23\right)\cdot 31^{2} + \left(13 a + 10\right)\cdot 31^{3} + \left(21 a + 27\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 29 + 17\cdot 31^{2} + 31^{3} + 24\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 5 a + 2 + \left(16 a + 4\right)\cdot 31 + 26\cdot 31^{2} + \left(14 a + 24\right)\cdot 31^{3} + \left(13 a + 24\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 26 a + 12 + 14 a\cdot 31 + \left(30 a + 11\right)\cdot 31^{2} + \left(16 a + 21\right)\cdot 31^{3} + \left(17 a + 6\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 14 + 16\cdot 31 + 10\cdot 31^{2} + 22\cdot 31^{3} + 5\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 27 a + 5 + \left(20 a + 16\right)\cdot 31 + \left(13 a + 17\right)\cdot 31^{2} + \left(17 a + 20\right)\cdot 31^{3} + \left(9 a + 25\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 6 + 24\cdot 31 + 17\cdot 31^{2} + 22\cdot 31^{3} + 9\cdot 31^{4} +O(31^{5})\)
|
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.