Basic invariants
Dimension: | $20$ |
Group: | $S_7$ |
Conductor: | \(495\!\cdots\!801\)\(\medspace = 371131^{10} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.1.371131.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 70 |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $S_7$ |
Projective stem field: | Galois closure of 7.1.371131.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - x^{4} - 2x^{2} - x - 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 223 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 223 }$: \( x^{2} + 221x + 3 \)
Roots:
$r_{ 1 }$ | $=$ | \( 10 a + 57 + \left(101 a + 4\right)\cdot 223 + 98\cdot 223^{2} + \left(91 a + 34\right)\cdot 223^{3} + \left(125 a + 89\right)\cdot 223^{4} +O(223^{5})\) |
$r_{ 2 }$ | $=$ | \( 174 a + 59 + \left(86 a + 158\right)\cdot 223 + \left(116 a + 140\right)\cdot 223^{2} + \left(25 a + 36\right)\cdot 223^{3} + \left(33 a + 162\right)\cdot 223^{4} +O(223^{5})\) |
$r_{ 3 }$ | $=$ | \( 83 + 56\cdot 223 + 204\cdot 223^{2} + 218\cdot 223^{3} + 116\cdot 223^{4} +O(223^{5})\) |
$r_{ 4 }$ | $=$ | \( 49 a + 184 + \left(136 a + 157\right)\cdot 223 + \left(106 a + 63\right)\cdot 223^{2} + \left(197 a + 194\right)\cdot 223^{3} + \left(189 a + 202\right)\cdot 223^{4} +O(223^{5})\) |
$r_{ 5 }$ | $=$ | \( 213 a + 77 + \left(121 a + 196\right)\cdot 223 + \left(222 a + 220\right)\cdot 223^{2} + \left(131 a + 215\right)\cdot 223^{3} + \left(97 a + 25\right)\cdot 223^{4} +O(223^{5})\) |
$r_{ 6 }$ | $=$ | \( 155 + 129\cdot 223 + 145\cdot 223^{2} + 45\cdot 223^{3} + 136\cdot 223^{4} +O(223^{5})\) |
$r_{ 7 }$ | $=$ | \( 54 + 189\cdot 223 + 18\cdot 223^{2} + 146\cdot 223^{3} + 158\cdot 223^{4} +O(223^{5})\) |
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 |
$1$ | $1$ | $()$ | $20$ |
$21$ | $2$ | $(1,2)$ | $0$ |
$105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
$105$ | $2$ | $(1,2)(3,4)$ | $-4$ |
$70$ | $3$ | $(1,2,3)$ | $2$ |
$280$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
$210$ | $4$ | $(1,2,3,4)$ | $0$ |
$630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
$504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
$210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $2$ |
$420$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
$840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
$720$ | $7$ | $(1,2,3,4,5,6,7)$ | $-1$ |
$504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
$420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.