Basic invariants
Dimension: | $20$ |
Group: | $S_7$ |
Conductor: | \(529\!\cdots\!249\)\(\medspace = 13^{10} \cdot 29^{10} \cdot 991^{10} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 7.1.373607.1 |
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 7.1.373607.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{7} - x^{6} + x^{5} - 2x^{3} + 3x^{2} - 2x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: \( x^{2} + 21x + 5 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 4 + 23^{2} + 22\cdot 23^{3} + 22\cdot 23^{4} +O(23^{5})\)
$r_{ 2 }$ |
$=$ |
\( 3 a + 16 + \left(2 a + 4\right)\cdot 23 + \left(18 a + 4\right)\cdot 23^{2} + \left(17 a + 4\right)\cdot 23^{3} + \left(10 a + 20\right)\cdot 23^{4} +O(23^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 20 a + 22 + \left(20 a + 5\right)\cdot 23 + \left(4 a + 15\right)\cdot 23^{2} + \left(5 a + 21\right)\cdot 23^{3} + 12 a\cdot 23^{4} +O(23^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 10 a + 15 + \left(12 a + 13\right)\cdot 23 + \left(a + 4\right)\cdot 23^{2} + \left(10 a + 13\right)\cdot 23^{3} + \left(8 a + 11\right)\cdot 23^{4} +O(23^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 13 a + 12 + \left(10 a + 5\right)\cdot 23 + \left(21 a + 18\right)\cdot 23^{2} + \left(12 a + 8\right)\cdot 23^{3} + \left(14 a + 18\right)\cdot 23^{4} +O(23^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 12 a + \left(21 a + 4\right)\cdot 23 + \left(13 a + 21\right)\cdot 23^{2} + \left(2 a + 3\right)\cdot 23^{3} + \left(7 a + 3\right)\cdot 23^{4} +O(23^{5})\)
| $r_{ 7 }$ |
$=$ |
\( 11 a + 1 + \left(a + 12\right)\cdot 23 + \left(9 a + 4\right)\cdot 23^{2} + \left(20 a + 18\right)\cdot 23^{3} + \left(15 a + 14\right)\cdot 23^{4} +O(23^{5})\)
| |
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.