Basic invariants
| Dimension: | $35$ |
| Group: | $S_7$ |
| Conductor: | \(542\!\cdots\!296\)\(\medspace = 2^{93} \cdot 3^{77} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 7.1.90699264.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 70 |
| Parity: | odd |
| Determinant: | 1.24.2t1.b.a |
| Projective image: | $S_7$ |
| Projective stem field: | Galois closure of 7.1.90699264.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{7} - x^{6} - x^{4} - 2x^{3} + 4x + 2 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 241 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 241 }$:
\( x^{2} + 238x + 7 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 112 + 211\cdot 241 + 35\cdot 241^{2} + 153\cdot 241^{3} + 146\cdot 241^{4} +O(241^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 182 a + 142 + \left(81 a + 104\right)\cdot 241 + \left(205 a + 89\right)\cdot 241^{2} + \left(169 a + 120\right)\cdot 241^{3} + \left(57 a + 104\right)\cdot 241^{4} +O(241^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 59 a + 206 + \left(159 a + 167\right)\cdot 241 + \left(35 a + 141\right)\cdot 241^{2} + \left(71 a + 183\right)\cdot 241^{3} + \left(183 a + 107\right)\cdot 241^{4} +O(241^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 29 + 44\cdot 241 + 168\cdot 241^{2} + 6\cdot 241^{3} + 183\cdot 241^{4} +O(241^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 193 a + 8 + \left(60 a + 50\right)\cdot 241 + 219\cdot 241^{2} + \left(226 a + 117\right)\cdot 241^{3} + \left(27 a + 135\right)\cdot 241^{4} +O(241^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 48 a + 105 + \left(180 a + 39\right)\cdot 241 + \left(240 a + 159\right)\cdot 241^{2} + \left(14 a + 72\right)\cdot 241^{3} + \left(213 a + 234\right)\cdot 241^{4} +O(241^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 122 + 105\cdot 241 + 150\cdot 241^{2} + 68\cdot 241^{3} + 52\cdot 241^{4} +O(241^{5})\)
|
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.