Basic invariants
| Dimension: | $20$ |
| Group: | $S_7$ |
| Conductor: | \(112\!\cdots\!201\)\(\medspace = 7^{12} \cdot 73^{10} \cdot 337^{10} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.3.1205449.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 70 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.3.1205449.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 2x^{6} + 4x^{4} - 4x^{3} + 3x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$:
\( x^{2} + 60x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 13 a + 54 + 46\cdot 61 + 26 a\cdot 61^{2} + \left(2 a + 11\right)\cdot 61^{3} + \left(9 a + 14\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 26 a + 48 + \left(26 a + 42\right)\cdot 61 + \left(55 a + 15\right)\cdot 61^{2} + \left(56 a + 14\right)\cdot 61^{3} + \left(5 a + 50\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 48 a + 6 + \left(60 a + 34\right)\cdot 61 + \left(34 a + 26\right)\cdot 61^{2} + \left(58 a + 48\right)\cdot 61^{3} + \left(51 a + 20\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 37 a + 19 + \left(26 a + 21\right)\cdot 61 + \left(45 a + 6\right)\cdot 61^{2} + 15 a\cdot 61^{3} + \left(12 a + 27\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 50 + 44\cdot 61 + 2\cdot 61^{2} + 61^{3} + 48\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 35 a + 13 + \left(34 a + 43\right)\cdot 61 + \left(5 a + 44\right)\cdot 61^{2} + \left(4 a + 15\right)\cdot 61^{3} + \left(55 a + 60\right)\cdot 61^{4} +O(61^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 24 a + 56 + \left(34 a + 10\right)\cdot 61 + \left(15 a + 25\right)\cdot 61^{2} + \left(45 a + 31\right)\cdot 61^{3} + \left(48 a + 23\right)\cdot 61^{4} +O(61^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $20$ |
| $21$ | $2$ | $(1,2)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-4$ |
| $70$ | $3$ | $(1,2,3)$ | $2$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
| $210$ | $4$ | $(1,2,3,4)$ | $0$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $2$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $-1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.