Basic invariants
| Dimension: | $35$ |
| Group: | $S_7$ |
| Conductor: | \(102\!\cdots\!576\)\(\medspace = 2^{102} \cdot 3^{74} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.1.241864704.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 126 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.1.241864704.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - 3x^{6} + 3x^{5} + x^{4} - 3x^{3} - 3x^{2} - 5x - 3 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 151 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 151 }$:
\( x^{2} + 149x + 6 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 102 + 81\cdot 151 + 14\cdot 151^{2} + 114\cdot 151^{3} + 106\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 66 a + 48 + \left(89 a + 135\right)\cdot 151 + \left(10 a + 61\right)\cdot 151^{2} + \left(123 a + 12\right)\cdot 151^{3} + \left(61 a + 86\right)\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 72 + 54\cdot 151 + 103\cdot 151^{2} + 147\cdot 151^{3} + 37\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 26 + 52\cdot 151 + 8\cdot 151^{2} + 43\cdot 151^{3} + 46\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 85 a + 29 + \left(61 a + 97\right)\cdot 151 + \left(140 a + 144\right)\cdot 151^{2} + \left(27 a + 96\right)\cdot 151^{3} + \left(89 a + 86\right)\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 83 + 58\cdot 151 + 116\cdot 151^{2} + 130\cdot 151^{3} + 14\cdot 151^{4} +O(151^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 96 + 124\cdot 151 + 3\cdot 151^{2} + 59\cdot 151^{3} + 74\cdot 151^{4} +O(151^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $35$ |
| $21$ | $2$ | $(1,2)$ | $-5$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $70$ | $3$ | $(1,2,3)$ | $-1$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $210$ | $4$ | $(1,2,3,4)$ | $1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.