Properties

Label 5.2e2_71_149.6t16.1
Dimension 5
Group $S_6$
Conductor $ 2^{2} \cdot 71 \cdot 149 $
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:$42316= 2^{2} \cdot 71 \cdot 149 $
Artin number field: Splitting field of $f= x^{6} - x^{3} + x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_6$
Parity: Odd

Galois action

Roots of defining polynomial

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} + 70 x + 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\left(73^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 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\left(73^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 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\left(73^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 16 + 31\cdot 73 + 18\cdot 73^{2} + 5\cdot 73^{3} + 20\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 58 + 30\cdot 73 + 24\cdot 73^{2} + 72\cdot 73^{3} + 60\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 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\left(73^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

Cycle notation
$(1,2)$
$(1,2,3,4,5,6)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character values
$c1$
$1$ $1$ $()$ $5$
$15$ $2$ $(1,2)(3,4)(5,6)$ $-1$
$15$ $2$ $(1,2)$ $3$
$45$ $2$ $(1,2)(3,4)$ $1$
$40$ $3$ $(1,2,3)(4,5,6)$ $-1$
$40$ $3$ $(1,2,3)$ $2$
$90$ $4$ $(1,2,3,4)(5,6)$ $-1$
$90$ $4$ $(1,2,3,4)$ $1$
$144$ $5$ $(1,2,3,4,5)$ $0$
$120$ $6$ $(1,2,3,4,5,6)$ $-1$
$120$ $6$ $(1,2,3)(4,5)$ $0$
The blue line marks the conjugacy class containing complex conjugation.