Basic invariants
Dimension: | $4$ |
Group: | $S_5$ |
Conductor: | \(22424\)\(\medspace = 2^{3} \cdot 2803 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.3.22424.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_5$ |
Parity: | odd |
Determinant: | 1.22424.2t1.b.a |
Projective image: | $S_5$ |
Projective stem field: | Galois closure of 5.3.22424.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - 2x^{3} - x^{2} - x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: \( x^{2} + 7x + 2 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 3 a + 3 + \left(10 a + 9\right)\cdot 11 + \left(2 a + 2\right)\cdot 11^{2} + \left(10 a + 4\right)\cdot 11^{3} + a\cdot 11^{4} +O(11^{5})\)
$r_{ 2 }$ |
$=$ |
\( 2 + 3\cdot 11 + 3\cdot 11^{2} + 9\cdot 11^{3} +O(11^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 8 a + 4 + 3\cdot 11 + \left(8 a + 4\right)\cdot 11^{2} + 9\cdot 11^{3} + \left(9 a + 8\right)\cdot 11^{4} +O(11^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 4 a + 4 + \left(a + 2\right)\cdot 11 + \left(9 a + 10\right)\cdot 11^{2} + \left(8 a + 2\right)\cdot 11^{3} + \left(a + 1\right)\cdot 11^{4} +O(11^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 7 a + 9 + \left(9 a + 3\right)\cdot 11 + \left(a + 1\right)\cdot 11^{2} + \left(2 a + 7\right)\cdot 11^{3} + \left(9 a + 10\right)\cdot 11^{4} +O(11^{5})\)
| |
Generators of the action on the roots $r_1, \ldots, r_{ 5 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 5 }$ | Character value |
$1$ | $1$ | $()$ | $4$ |
$10$ | $2$ | $(1,2)$ | $2$ |
$15$ | $2$ | $(1,2)(3,4)$ | $0$ |
$20$ | $3$ | $(1,2,3)$ | $1$ |
$30$ | $4$ | $(1,2,3,4)$ | $0$ |
$24$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
$20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.