Basic invariants
Dimension: | $16$ |
Group: | $S_6$ |
Conductor: | \(409\!\cdots\!336\)\(\medspace = 2^{16} \cdot 83^{8} \cdot 479^{8} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.0.159028.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 36T1252 |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $S_6$ |
Projective stem field: | Galois closure of 6.0.159028.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - x^{5} + 2x^{4} - 2x^{3} + 2x^{2} + x + 1 \) . |
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:
$r_{ 1 }$ | $=$ | \( 31 a + 42 + \left(11 a + 48\right)\cdot 73 + \left(30 a + 24\right)\cdot 73^{2} + \left(59 a + 27\right)\cdot 73^{3} + \left(45 a + 46\right)\cdot 73^{4} +O(73^{5})\) |
$r_{ 2 }$ | $=$ | \( 42 a + 62 + \left(61 a + 51\right)\cdot 73 + \left(42 a + 30\right)\cdot 73^{2} + \left(13 a + 29\right)\cdot 73^{3} + \left(27 a + 51\right)\cdot 73^{4} +O(73^{5})\) |
$r_{ 3 }$ | $=$ | \( 8 a + 2 + \left(23 a + 46\right)\cdot 73 + \left(35 a + 25\right)\cdot 73^{2} + \left(11 a + 69\right)\cdot 73^{3} + \left(8 a + 38\right)\cdot 73^{4} +O(73^{5})\) |
$r_{ 4 }$ | $=$ | \( 43 a + 16 + 2 a\cdot 73 + \left(40 a + 29\right)\cdot 73^{2} + \left(23 a + 33\right)\cdot 73^{3} + \left(54 a + 18\right)\cdot 73^{4} +O(73^{5})\) |
$r_{ 5 }$ | $=$ | \( 30 a + 72 + \left(70 a + 37\right)\cdot 73 + 32 a\cdot 73^{2} + \left(49 a + 64\right)\cdot 73^{3} + \left(18 a + 11\right)\cdot 73^{4} +O(73^{5})\) |
$r_{ 6 }$ | $=$ | \( 65 a + 26 + \left(49 a + 34\right)\cdot 73 + \left(37 a + 35\right)\cdot 73^{2} + \left(61 a + 68\right)\cdot 73^{3} + \left(64 a + 51\right)\cdot 73^{4} +O(73^{5})\) |
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 |
$1$ | $1$ | $()$ | $16$ |
$15$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
$15$ | $2$ | $(1,2)$ | $0$ |
$45$ | $2$ | $(1,2)(3,4)$ | $0$ |
$40$ | $3$ | $(1,2,3)(4,5,6)$ | $-2$ |
$40$ | $3$ | $(1,2,3)$ | $-2$ |
$90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
$90$ | $4$ | $(1,2,3,4)$ | $0$ |
$144$ | $5$ | $(1,2,3,4,5)$ | $1$ |
$120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
$120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.