Basic invariants
| Dimension: | $15$ |
| Group: | $S_7$ |
| Conductor: | \(226\!\cdots\!624\)\(\medspace = 2^{40} \cdot 3^{30} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.1.60466176.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 42T412 |
| Parity: | odd |
| Determinant: | 1.4.2t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.1.60466176.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 2x^{6} + 3x^{5} + x^{4} - 5x^{3} + 12x^{2} - 7x + 5 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$:
\( x^{3} + x + 14 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 8\cdot 17 + 14\cdot 17^{2} + 17^{3} + 11\cdot 17^{4} +O(17^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 16 a^{2} + 15 + \left(14 a^{2} + 11 a + 7\right)\cdot 17 + \left(3 a^{2} + 15 a + 16\right)\cdot 17^{2} + \left(2 a^{2} + 2\right)\cdot 17^{3} + \left(14 a^{2} + 16 a + 5\right)\cdot 17^{4} +O(17^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 12 a^{2} + 7 a + 9 + \left(3 a^{2} + 10 a + 7\right)\cdot 17 + \left(15 a^{2} + 7 a + 8\right)\cdot 17^{2} + \left(4 a^{2} + 12 a + 12\right)\cdot 17^{3} + \left(16 a^{2} + 15 a + 5\right)\cdot 17^{4} +O(17^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 10 a^{2} + 11 a + 2 + \left(15 a^{2} + 7 a + 4\right)\cdot 17 + \left(2 a^{2} + 3 a\right)\cdot 17^{2} + \left(13 a^{2} + 15 a + 1\right)\cdot 17^{3} + \left(11 a^{2} + 14\right)\cdot 17^{4} +O(17^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 8 a^{2} + 7 a + 4 + \left(15 a^{2} + 11 a + 8\right)\cdot 17 + \left(7 a^{2} + 2 a + 13\right)\cdot 17^{2} + \left(7 a^{2} + 16 a\right)\cdot 17^{3} + \left(14 a^{2} + 11\right)\cdot 17^{4} +O(17^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 10 a^{2} + 10 a + 11 + \left(3 a^{2} + 11 a + 11\right)\cdot 17 + \left(5 a^{2} + 15 a + 11\right)\cdot 17^{2} + \left(7 a^{2} + 16 a\right)\cdot 17^{3} + \left(5 a^{2} + 16 a + 5\right)\cdot 17^{4} +O(17^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 12 a^{2} + 16 a + 9 + \left(14 a^{2} + 15 a + 3\right)\cdot 17 + \left(15 a^{2} + 5 a + 3\right)\cdot 17^{2} + \left(15 a^{2} + 6 a + 14\right)\cdot 17^{3} + \left(5 a^{2} + 15\right)\cdot 17^{4} +O(17^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $15$ |
| $21$ | $2$ | $(1,2)$ | $5$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $70$ | $3$ | $(1,2,3)$ | $3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.