Basic invariants
Dimension: | $5$ |
Group: | $S_6$ |
Conductor: | \(31882152733829\)\(\medspace = 37^{3} \cdot 857^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.2.31709.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_6$ |
Parity: | even |
Determinant: | 1.31709.2t1.a.a |
Projective image: | $S_6$ |
Projective stem field: | Galois closure of 6.2.31709.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - x^{5} + 2x^{3} - 2x^{2} + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$: \( x^{2} + 70x + 5 \)
Roots:
$r_{ 1 }$ | $=$ | \( 39 a + 28 + \left(49 a + 25\right)\cdot 73 + \left(2 a + 71\right)\cdot 73^{2} + \left(48 a + 20\right)\cdot 73^{3} + \left(55 a + 50\right)\cdot 73^{4} +O(73^{5})\) |
$r_{ 2 }$ | $=$ | \( 40 a + 28 + \left(67 a + 43\right)\cdot 73 + 37\cdot 73^{2} + \left(38 a + 30\right)\cdot 73^{3} + \left(49 a + 25\right)\cdot 73^{4} +O(73^{5})\) |
$r_{ 3 }$ | $=$ | \( 34 a + 72 + \left(23 a + 61\right)\cdot 73 + \left(70 a + 29\right)\cdot 73^{2} + \left(24 a + 16\right)\cdot 73^{3} + \left(17 a + 23\right)\cdot 73^{4} +O(73^{5})\) |
$r_{ 4 }$ | $=$ | \( 71 + 6\cdot 73 + 42\cdot 73^{2} + 36\cdot 73^{3} + 34\cdot 73^{4} +O(73^{5})\) |
$r_{ 5 }$ | $=$ | \( 33 a + 2 + \left(5 a + 60\right)\cdot 73 + \left(72 a + 45\right)\cdot 73^{2} + \left(34 a + 70\right)\cdot 73^{3} + \left(23 a + 62\right)\cdot 73^{4} +O(73^{5})\) |
$r_{ 6 }$ | $=$ | \( 19 + 21\cdot 73 + 65\cdot 73^{2} + 43\cdot 73^{3} + 22\cdot 73^{4} +O(73^{5})\) |
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 |
$1$ | $1$ | $()$ | $5$ |
$15$ | $2$ | $(1,2)(3,4)(5,6)$ | $3$ |
$15$ | $2$ | $(1,2)$ | $-1$ |
$45$ | $2$ | $(1,2)(3,4)$ | $1$ |
$40$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
$40$ | $3$ | $(1,2,3)$ | $-1$ |
$90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
$90$ | $4$ | $(1,2,3,4)$ | $1$ |
$144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
$120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
$120$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.