Basic invariants
Dimension: | $4$ |
Group: | $S_5$ |
Conductor: | \(1421171965907\)\(\medspace = 11243^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.3.11243.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_5$ |
Parity: | odd |
Determinant: | 1.11243.2t1.a.a |
Projective image: | $S_5$ |
Projective stem field: | Galois closure of 5.3.11243.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - x^{4} - 2x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: \( x^{2} + 29x + 3 \)
Roots:
$r_{ 1 }$ | $=$ | \( 27 + 10\cdot 31 + 15\cdot 31^{2} + 9\cdot 31^{3} + 19\cdot 31^{4} +O(31^{5})\) |
$r_{ 2 }$ | $=$ | \( 14 a + 18 + \left(14 a + 12\right)\cdot 31 + \left(8 a + 10\right)\cdot 31^{2} + 17\cdot 31^{3} + \left(28 a + 24\right)\cdot 31^{4} +O(31^{5})\) |
$r_{ 3 }$ | $=$ | \( 17 a + 15 + \left(16 a + 27\right)\cdot 31 + \left(22 a + 12\right)\cdot 31^{2} + \left(30 a + 9\right)\cdot 31^{3} + \left(2 a + 18\right)\cdot 31^{4} +O(31^{5})\) |
$r_{ 4 }$ | $=$ | \( 21 a + 27 + 30\cdot 31 + \left(16 a + 26\right)\cdot 31^{2} + \left(27 a + 8\right)\cdot 31^{3} + \left(20 a + 8\right)\cdot 31^{4} +O(31^{5})\) |
$r_{ 5 }$ | $=$ | \( 10 a + 7 + \left(30 a + 11\right)\cdot 31 + \left(14 a + 27\right)\cdot 31^{2} + \left(3 a + 16\right)\cdot 31^{3} + \left(10 a + 22\right)\cdot 31^{4} +O(31^{5})\) |
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.