Basic invariants
Dimension: | $6$ |
Group: | $S_5$ |
Conductor: | \(898254927\)\(\medspace = 3^{4} \cdot 223^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.3.18063.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 20T30 |
Parity: | odd |
Determinant: | 1.223.2t1.a.a |
Projective image: | $S_5$ |
Projective stem field: | Galois closure of 5.3.18063.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - x^{4} + x^{3} - 2x^{2} - x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: \( x^{2} + 33x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( 5 a + 33 + \left(22 a + 8\right)\cdot 37 + \left(21 a + 6\right)\cdot 37^{2} + \left(19 a + 25\right)\cdot 37^{3} + \left(35 a + 36\right)\cdot 37^{4} +O(37^{5})\) |
$r_{ 2 }$ | $=$ | \( 32 a + 11 + 3 a\cdot 37 + \left(7 a + 17\right)\cdot 37^{2} + \left(30 a + 36\right)\cdot 37^{3} + \left(32 a + 16\right)\cdot 37^{4} +O(37^{5})\) |
$r_{ 3 }$ | $=$ | \( 5 a + 28 + \left(33 a + 20\right)\cdot 37 + \left(29 a + 4\right)\cdot 37^{2} + \left(6 a + 2\right)\cdot 37^{3} + \left(4 a + 7\right)\cdot 37^{4} +O(37^{5})\) |
$r_{ 4 }$ | $=$ | \( 32 a + 16 + \left(14 a + 18\right)\cdot 37 + \left(15 a + 33\right)\cdot 37^{2} + \left(17 a + 7\right)\cdot 37^{3} + \left(a + 11\right)\cdot 37^{4} +O(37^{5})\) |
$r_{ 5 }$ | $=$ | \( 24 + 25\cdot 37 + 12\cdot 37^{2} + 2\cdot 37^{3} + 2\cdot 37^{4} +O(37^{5})\) |
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 |
$1$ | $1$ | $()$ | $6$ |
$10$ | $2$ | $(1,2)$ | $0$ |
$15$ | $2$ | $(1,2)(3,4)$ | $-2$ |
$20$ | $3$ | $(1,2,3)$ | $0$ |
$30$ | $4$ | $(1,2,3,4)$ | $0$ |
$24$ | $5$ | $(1,2,3,4,5)$ | $1$ |
$20$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.