Basic invariants
Dimension: | $21$ |
Group: | $S_7$ |
Conductor: | \(574\!\cdots\!896\)\(\medspace = 2^{56} \cdot 3^{48} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.1.90699264.2 |
Galois orbit size: | $1$ |
Smallest permutation container: | 84 |
Parity: | even |
Determinant: | 1.1.1t1.a.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 |
$1$ | $1$ | $()$ | $21$ |
$21$ | $2$ | $(1,2)$ | $1$ |
$105$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
$105$ | $2$ | $(1,2)(3,4)$ | $1$ |
$70$ | $3$ | $(1,2,3)$ | $-3$ |
$280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
$210$ | $4$ | $(1,2,3,4)$ | $-1$ |
$630$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
$504$ | $5$ | $(1,2,3,4,5)$ | $1$ |
$210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $1$ |
$420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
$840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
$720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
$504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $1$ |
$420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.