Basic invariants
Dimension: | $6$ |
Group: | $S_5$ |
Conductor: | \(345740920830625\)\(\medspace = 5^{4} \cdot 8209^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.5.205225.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 20T30 |
Parity: | even |
Determinant: | 1.8209.2t1.a.a |
Projective image: | $S_5$ |
Projective stem field: | Galois closure of 5.5.205225.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - x^{4} - 6x^{3} + 3x^{2} + 7x - 3 \) . |
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:
$r_{ 1 }$ | $=$ | \( 10 a + 32 + \left(26 a + 4\right)\cdot 37 + \left(6 a + 6\right)\cdot 37^{2} + \left(31 a + 8\right)\cdot 37^{3} + 35 a\cdot 37^{4} +O(37^{5})\) |
$r_{ 2 }$ | $=$ | \( 27 a + 35 + \left(10 a + 25\right)\cdot 37 + \left(30 a + 6\right)\cdot 37^{2} + \left(5 a + 15\right)\cdot 37^{3} + \left(a + 1\right)\cdot 37^{4} +O(37^{5})\) |
$r_{ 3 }$ | $=$ | \( 20 + 21\cdot 37 + 21\cdot 37^{2} + 14\cdot 37^{3} + 27\cdot 37^{4} +O(37^{5})\) |
$r_{ 4 }$ | $=$ | \( 11 + 9\cdot 37 + 26\cdot 37^{2} + 12\cdot 37^{3} + 30\cdot 37^{4} +O(37^{5})\) |
$r_{ 5 }$ | $=$ | \( 14 + 12\cdot 37 + 13\cdot 37^{2} + 23\cdot 37^{3} + 14\cdot 37^{4} +O(37^{5})\) |
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.