Properties

Label 5.2e12_3e10.6t15.2
Dimension 5
Group $A_6$
Conductor $ 2^{12} \cdot 3^{10}$
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$5$
Group:$A_6$
Conductor:$241864704= 2^{12} \cdot 3^{10} $
Artin number field: Splitting field of $f= x^{6} + 6 x^{4} - 4 x^{3} - 3 x^{2} - 12 x - 12 $ 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 $\Q_{ 53 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $ x^{2} + 49 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 31 + 53 + 45\cdot 53^{2} + 49\cdot 53^{3} + 17\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 19 a + 40 + \left(14 a + 14\right)\cdot 53 + 40 a\cdot 53^{2} + \left(40 a + 38\right)\cdot 53^{3} + \left(7 a + 16\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 14 a + 23 + \left(49 a + 12\right)\cdot 53 + \left(22 a + 1\right)\cdot 53^{2} + \left(20 a + 9\right)\cdot 53^{3} + \left(32 a + 18\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 29 + 40\cdot 53 + 27\cdot 53^{2} + 45\cdot 53^{3} + 24\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 34 a + 10 + 38 a\cdot 53 + \left(12 a + 41\right)\cdot 53^{2} + \left(12 a + 1\right)\cdot 53^{3} + \left(45 a + 7\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 39 a + 26 + \left(3 a + 36\right)\cdot 53 + \left(30 a + 43\right)\cdot 53^{2} + \left(32 a + 14\right)\cdot 53^{3} + \left(20 a + 21\right)\cdot 53^{4} +O\left(53^{ 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
$c1$
$1$ $1$ $()$ $5$
$45$ $2$ $(1,2)(3,4)$ $1$
$40$ $3$ $(1,2,3)(4,5,6)$ $2$
$40$ $3$ $(1,2,3)$ $-1$
$90$ $4$ $(1,2,3,4)(5,6)$ $-1$
$72$ $5$ $(1,2,3,4,5)$ $0$
$72$ $5$ $(1,3,4,5,2)$ $0$
The blue line marks the conjugacy class containing complex conjugation.