Basic invariants
Dimension: | $5$ |
Group: | $S_5$ |
Conductor: | \(4182697396374561\)\(\medspace = 3^{3} \cdot 43^{3} \cdot 1249^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.5.161121.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $\PGL(2,5)$ |
Parity: | even |
Determinant: | 1.161121.2t1.a.a |
Projective image: | $S_5$ |
Projective stem field: | Galois closure of 5.5.161121.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - x^{4} - 6x^{3} + 3x^{2} + 5x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: \( x^{2} + 38x + 6 \)
Roots:
$r_{ 1 }$ | $=$ | \( 2 a + 5 + \left(3 a + 23\right)\cdot 41 + \left(39 a + 5\right)\cdot 41^{2} + \left(24 a + 24\right)\cdot 41^{3} + \left(9 a + 40\right)\cdot 41^{4} +O(41^{5})\) |
$r_{ 2 }$ | $=$ | \( 39 a + 11 + \left(37 a + 30\right)\cdot 41 + \left(a + 37\right)\cdot 41^{2} + \left(16 a + 18\right)\cdot 41^{3} + \left(31 a + 3\right)\cdot 41^{4} +O(41^{5})\) |
$r_{ 3 }$ | $=$ | \( 19 a + 18 + \left(17 a + 28\right)\cdot 41 + \left(28 a + 1\right)\cdot 41^{2} + \left(35 a + 40\right)\cdot 41^{3} + \left(2 a + 17\right)\cdot 41^{4} +O(41^{5})\) |
$r_{ 4 }$ | $=$ | \( 15 + 20\cdot 41 + 8\cdot 41^{2} + 3\cdot 41^{3} + 29\cdot 41^{4} +O(41^{5})\) |
$r_{ 5 }$ | $=$ | \( 22 a + 34 + \left(23 a + 20\right)\cdot 41 + \left(12 a + 28\right)\cdot 41^{2} + \left(5 a + 36\right)\cdot 41^{3} + \left(38 a + 31\right)\cdot 41^{4} +O(41^{5})\) |
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 |
$1$ | $1$ | $()$ | $5$ |
$10$ | $2$ | $(1,2)$ | $-1$ |
$15$ | $2$ | $(1,2)(3,4)$ | $1$ |
$20$ | $3$ | $(1,2,3)$ | $-1$ |
$30$ | $4$ | $(1,2,3,4)$ | $1$ |
$24$ | $5$ | $(1,2,3,4,5)$ | $0$ |
$20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.