Basic invariants
Dimension: | $5$ |
Group: | $S_6$ |
Conductor: | \(241856\)\(\medspace = 2^{6} \cdot 3779 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.2.241856.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_6$ |
Parity: | even |
Determinant: | 1.15116.2t1.a.a |
Projective image: | $S_6$ |
Projective stem field: | Galois closure of 6.2.241856.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - 2x^{5} + 2x^{4} - 4x^{3} + 5x^{2} - 4x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$: \( x^{2} + 69x + 7 \)
Roots:
$r_{ 1 }$ | $=$ | \( 33 a + 19 + \left(19 a + 7\right)\cdot 71 + \left(39 a + 50\right)\cdot 71^{2} + \left(8 a + 40\right)\cdot 71^{3} + \left(47 a + 37\right)\cdot 71^{4} +O(71^{5})\) |
$r_{ 2 }$ | $=$ | \( 66 a + 9 + \left(42 a + 66\right)\cdot 71 + \left(48 a + 11\right)\cdot 71^{2} + \left(45 a + 44\right)\cdot 71^{3} + \left(25 a + 44\right)\cdot 71^{4} +O(71^{5})\) |
$r_{ 3 }$ | $=$ | \( 38 a + 14 + \left(51 a + 13\right)\cdot 71 + \left(31 a + 38\right)\cdot 71^{2} + \left(62 a + 18\right)\cdot 71^{3} + \left(23 a + 52\right)\cdot 71^{4} +O(71^{5})\) |
$r_{ 4 }$ | $=$ | \( 5 a + 70 + \left(28 a + 14\right)\cdot 71 + \left(22 a + 66\right)\cdot 71^{2} + \left(25 a + 15\right)\cdot 71^{3} + \left(45 a + 50\right)\cdot 71^{4} +O(71^{5})\) |
$r_{ 5 }$ | $=$ | \( 12 + 15\cdot 71 + 70\cdot 71^{2} + 21\cdot 71^{3} + 49\cdot 71^{4} +O(71^{5})\) |
$r_{ 6 }$ | $=$ | \( 20 + 25\cdot 71 + 47\cdot 71^{2} + 50\cdot 71^{4} +O(71^{5})\) |
Generators of the action on the roots $r_1, \ldots, r_{ 6 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 6 }$ | Character value |
$1$ | $1$ | $()$ | $5$ |
$15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
$15$ | $2$ | $(1,2)$ | $3$ |
$45$ | $2$ | $(1,2)(3,4)$ | $1$ |
$40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
$40$ | $3$ | $(1,2,3)$ | $2$ |
$90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
$90$ | $4$ | $(1,2,3,4)$ | $1$ |
$144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
$120$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
$120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.