Basic invariants
| Dimension: | $20$ |
| Group: | $S_7$ |
| Conductor: | \(729\!\cdots\!401\)\(\medspace = 397^{10} \cdot 153997^{10} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.7.61136809.1 |
| 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 7.7.61136809.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 7x^{5} - 2x^{4} + 12x^{3} + 5x^{2} - 3x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 109 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 109 }$:
\( x^{2} + 108x + 6 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 58 a + 14 + \left(73 a + 105\right)\cdot 109 + \left(2 a + 81\right)\cdot 109^{2} + \left(74 a + 49\right)\cdot 109^{3} + \left(58 a + 57\right)\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 105 a + 45 + \left(23 a + 48\right)\cdot 109 + \left(2 a + 64\right)\cdot 109^{2} + \left(85 a + 18\right)\cdot 109^{3} + \left(54 a + 19\right)\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 4 a + 41 + \left(85 a + 76\right)\cdot 109 + \left(106 a + 42\right)\cdot 109^{2} + \left(23 a + 101\right)\cdot 109^{3} + \left(54 a + 97\right)\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 100 + 13\cdot 109 + 8\cdot 109^{2} + 36\cdot 109^{3} + 16\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 94 a + 35 + \left(48 a + 58\right)\cdot 109 + \left(44 a + 61\right)\cdot 109^{2} + \left(101 a + 80\right)\cdot 109^{3} + \left(32 a + 26\right)\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 51 a + 72 + \left(35 a + 11\right)\cdot 109 + \left(106 a + 11\right)\cdot 109^{2} + \left(34 a + 12\right)\cdot 109^{3} + \left(50 a + 42\right)\cdot 109^{4} +O(109^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 15 a + 20 + \left(60 a + 13\right)\cdot 109 + \left(64 a + 57\right)\cdot 109^{2} + \left(7 a + 28\right)\cdot 109^{3} + \left(76 a + 67\right)\cdot 109^{4} +O(109^{5})\)
|
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.