Basic invariants
Dimension: | $5$ |
Group: | $S_6$ |
Conductor: | \(42316\)\(\medspace = 2^{2} \cdot 71 \cdot 149 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 6.0.42316.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_6$ |
Parity: | odd |
Determinant: | 1.10579.2t1.a.a |
Projective image: | $S_6$ |
Projective stem field: | Galois closure of 6.0.42316.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - x^{3} + x + 1 \) . |
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:
$r_{ 1 }$ | $=$ |
\( 12 a + 66 + \left(16 a + 31\right)\cdot 73 + \left(44 a + 20\right)\cdot 73^{2} + \left(34 a + 60\right)\cdot 73^{3} + \left(72 a + 71\right)\cdot 73^{4} +O(73^{5})\)
$r_{ 2 }$ |
$=$ |
\( 70 a + 66 + \left(6 a + 52\right)\cdot 73 + \left(55 a + 39\right)\cdot 73^{2} + \left(47 a + 9\right)\cdot 73^{3} + \left(50 a + 36\right)\cdot 73^{4} +O(73^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 3 a + 57 + \left(66 a + 3\right)\cdot 73 + \left(17 a + 52\right)\cdot 73^{2} + \left(25 a + 24\right)\cdot 73^{3} + \left(22 a + 67\right)\cdot 73^{4} +O(73^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 16 + 31\cdot 73 + 18\cdot 73^{2} + 5\cdot 73^{3} + 20\cdot 73^{4} +O(73^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 58 + 30\cdot 73 + 24\cdot 73^{2} + 72\cdot 73^{3} + 60\cdot 73^{4} +O(73^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 61 a + 29 + \left(56 a + 68\right)\cdot 73 + \left(28 a + 63\right)\cdot 73^{2} + \left(38 a + 46\right)\cdot 73^{3} + 35\cdot 73^{4} +O(73^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $5$ |
$15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
$15$ | $2$ | $(1,2)$ | $3$ |
$45$ | $2$ | $(1,2)(3,4)$ | $1$ |
$40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
$40$ | $3$ | $(1,2,3)$ | $2$ |
$90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
$90$ | $4$ | $(1,2,3,4)$ | $1$ |
$144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
$120$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
$120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.