Basic invariants
Dimension: | $5$ |
Group: | $S_5$ |
Conductor: | \(77363231581\)\(\medspace = 4261^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.1.4261.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $\PGL(2,5)$ |
Parity: | even |
Determinant: | 1.4261.2t1.a.a |
Projective image: | $S_5$ |
Projective stem field: | Galois closure of 5.1.4261.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - 2x^{2} - 2x - 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: \( x^{2} + 16x + 3 \)
Roots:
$r_{ 1 }$ | $=$ | \( 11 + 12\cdot 17 + 2\cdot 17^{2} + 12\cdot 17^{3} + 14\cdot 17^{4} +O(17^{5})\) |
$r_{ 2 }$ | $=$ | \( 6 a + 1 + \left(4 a + 8\right)\cdot 17 + \left(10 a + 10\right)\cdot 17^{2} + \left(11 a + 4\right)\cdot 17^{3} + \left(a + 16\right)\cdot 17^{4} +O(17^{5})\) |
$r_{ 3 }$ | $=$ | \( 8 + 3\cdot 17 + 15\cdot 17^{2} + 11\cdot 17^{3} + 4\cdot 17^{4} +O(17^{5})\) |
$r_{ 4 }$ | $=$ | \( 7 + 3\cdot 17 + 6\cdot 17^{2} + 16\cdot 17^{3} + 8\cdot 17^{4} +O(17^{5})\) |
$r_{ 5 }$ | $=$ | \( 11 a + 7 + \left(12 a + 6\right)\cdot 17 + \left(6 a + 16\right)\cdot 17^{2} + \left(5 a + 5\right)\cdot 17^{3} + \left(15 a + 6\right)\cdot 17^{4} +O(17^{5})\) |
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.