Basic invariants
| Dimension: | $35$ |
| Group: | $S_7$ |
| Conductor: | \(173\!\cdots\!472\)\(\medspace = 2^{98} \cdot 3^{77} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.1.725594112.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 70 |
| Parity: | odd |
| Determinant: | 1.3.2t1.a.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.1.725594112.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - x^{6} + 18x^{3} - 18x^{2} - 24x - 48 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 149 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 149 }$:
\( x^{2} + 145x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 8 + 27\cdot 149 + 66\cdot 149^{2} + 34\cdot 149^{3} + 110\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 8 a + 38 + \left(106 a + 141\right)\cdot 149 + \left(83 a + 75\right)\cdot 149^{2} + \left(3 a + 109\right)\cdot 149^{3} + \left(87 a + 31\right)\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 141 a + 70 + \left(42 a + 110\right)\cdot 149 + \left(65 a + 6\right)\cdot 149^{2} + \left(145 a + 40\right)\cdot 149^{3} + \left(61 a + 78\right)\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 97 a + 21 + \left(64 a + 91\right)\cdot 149 + \left(42 a + 14\right)\cdot 149^{2} + \left(30 a + 144\right)\cdot 149^{3} + \left(80 a + 148\right)\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 52 a + 111 + \left(84 a + 103\right)\cdot 149 + \left(106 a + 119\right)\cdot 149^{2} + \left(118 a + 73\right)\cdot 149^{3} + \left(68 a + 141\right)\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 4 a + 92 + \left(21 a + 95\right)\cdot 149 + \left(59 a + 48\right)\cdot 149^{2} + \left(97 a + 6\right)\cdot 149^{3} + \left(61 a + 117\right)\cdot 149^{4} +O(149^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 145 a + 108 + \left(127 a + 26\right)\cdot 149 + \left(89 a + 115\right)\cdot 149^{2} + \left(51 a + 38\right)\cdot 149^{3} + \left(87 a + 117\right)\cdot 149^{4} +O(149^{5})\)
|
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.