Properties

Label 10.3e20_7e8.30t88.2
Dimension 10
Group $A_6$
Conductor $ 3^{20} \cdot 7^{8}$
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$10$
Group:$A_6$
Conductor:$20100618201669201= 3^{20} \cdot 7^{8} $
Artin number field: Splitting field of $f= x^{6} - 3 x^{5} + 3 x^{4} + 9 x^{3} - 18 x^{2} - 9 x + 18 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $A_6$
Parity: Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 151 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 151 }$: $ x^{2} + 149 x + 6 $
Roots:
$r_{ 1 }$ $=$ $ 3 a + 43 + \left(58 a + 148\right)\cdot 151 + \left(49 a + 52\right)\cdot 151^{2} + \left(129 a + 106\right)\cdot 151^{3} + \left(108 a + 148\right)\cdot 151^{4} +O\left(151^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 48 a + 76 + \left(8 a + 37\right)\cdot 151 + \left(104 a + 23\right)\cdot 151^{2} + \left(108 a + 76\right)\cdot 151^{3} + \left(150 a + 133\right)\cdot 151^{4} +O\left(151^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 148 a + 49 + \left(92 a + 110\right)\cdot 151 + \left(101 a + 93\right)\cdot 151^{2} + \left(21 a + 13\right)\cdot 151^{3} + \left(42 a + 86\right)\cdot 151^{4} +O\left(151^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 123 + 50\cdot 151 + 86\cdot 151^{2} + 128\cdot 151^{3} + 24\cdot 151^{4} +O\left(151^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 103 a + 21 + \left(142 a + 6\right)\cdot 151 + \left(46 a + 72\right)\cdot 151^{2} + \left(42 a + 38\right)\cdot 151^{3} + 24\cdot 151^{4} +O\left(151^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 144 + 99\cdot 151 + 124\cdot 151^{2} + 89\cdot 151^{3} + 35\cdot 151^{4} +O\left(151^{ 5 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character values
$c1$
$1$ $1$ $()$ $10$
$45$ $2$ $(1,2)(3,4)$ $-2$
$40$ $3$ $(1,2,3)(4,5,6)$ $1$
$40$ $3$ $(1,2,3)$ $1$
$90$ $4$ $(1,2,3,4)(5,6)$ $0$
$72$ $5$ $(1,2,3,4,5)$ $0$
$72$ $5$ $(1,3,4,5,2)$ $0$
The blue line marks the conjugacy class containing complex conjugation.