Basic invariants
Dimension: | $5$ |
Group: | $S_5$ |
Conductor: | \(22412680708304\)\(\medspace = 2^{4} \cdot 67^{3} \cdot 167^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.5.179024.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $\PGL(2,5)$ |
Parity: | even |
Determinant: | 1.11189.2t1.a.a |
Projective image: | $S_5$ |
Projective stem field: | Galois closure of 5.5.179024.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - 8x^{3} + 6x - 2 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: \( x^{2} + 24x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( 23 a + 23 + \left(20 a + 13\right)\cdot 29 + \left(5 a + 1\right)\cdot 29^{2} + \left(14 a + 9\right)\cdot 29^{3} + \left(16 a + 2\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 2 }$ | $=$ | \( 15 a + 17 + \left(21 a + 4\right)\cdot 29 + \left(25 a + 2\right)\cdot 29^{2} + \left(14 a + 22\right)\cdot 29^{3} + \left(21 a + 11\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 3 }$ | $=$ | \( 6 a + 22 + \left(8 a + 7\right)\cdot 29 + \left(23 a + 9\right)\cdot 29^{2} + \left(14 a + 16\right)\cdot 29^{3} + \left(12 a + 12\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 4 }$ | $=$ | \( 14 a + 5 + \left(7 a + 10\right)\cdot 29 + \left(3 a + 22\right)\cdot 29^{2} + \left(14 a + 12\right)\cdot 29^{3} + \left(7 a + 17\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 5 }$ | $=$ | \( 20 + 21\cdot 29 + 22\cdot 29^{2} + 26\cdot 29^{3} + 13\cdot 29^{4} +O(29^{5})\) |
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.