Basic invariants
| Dimension: | $14$ |
| Group: | $S_7$ |
| Conductor: | \(150\!\cdots\!601\)\(\medspace = 11^{10} \cdot 41^{10} \cdot 461^{10} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.1.207911.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 42T413 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.1.207911.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - x^{5} - x^{4} - x^{3} + x^{2} + x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 157 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 157 }$:
\( x^{2} + 152x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 44 a + 152 + \left(119 a + 131\right)\cdot 157 + \left(34 a + 89\right)\cdot 157^{2} + \left(123 a + 101\right)\cdot 157^{3} + \left(112 a + 63\right)\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 53 a + 74 + \left(62 a + 109\right)\cdot 157 + \left(5 a + 122\right)\cdot 157^{2} + \left(80 a + 49\right)\cdot 157^{3} + \left(81 a + 67\right)\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 134 a + 31 + \left(82 a + 25\right)\cdot 157 + \left(152 a + 20\right)\cdot 157^{2} + \left(124 a + 156\right)\cdot 157^{3} + \left(133 a + 125\right)\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 104 a + 25 + \left(94 a + 54\right)\cdot 157 + \left(151 a + 87\right)\cdot 157^{2} + \left(76 a + 130\right)\cdot 157^{3} + \left(75 a + 80\right)\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 58 + 102\cdot 157 + 91\cdot 157^{2} + 134\cdot 157^{3} + 57\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 23 a + 73 + \left(74 a + 148\right)\cdot 157 + \left(4 a + 71\right)\cdot 157^{2} + 32 a\cdot 157^{3} + \left(23 a + 42\right)\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 113 a + 58 + \left(37 a + 56\right)\cdot 157 + \left(122 a + 144\right)\cdot 157^{2} + \left(33 a + 54\right)\cdot 157^{3} + \left(44 a + 33\right)\cdot 157^{4} +O(157^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $14$ |
| $21$ | $2$ | $(1,2)$ | $-6$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $2$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $210$ | $4$ | $(1,2,3,4)$ | $0$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $2$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $1$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $-1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.