Properties

Label 5.7e4_283e2.6t15.1
Dimension 5
Group $A_6$
Conductor $ 7^{4} \cdot 283^{2}$
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$5$
Group:$A_6$
Conductor:$192293689= 7^{4} \cdot 283^{2} $
Artin number field: Splitting field of $f= x^{6} - x^{5} - 12 x^{4} + 7 x^{3} + 21 x^{2} - 7 x - 7 $ 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:
$r_{ 1 }$ $=$ $ 32 + 52\cdot 53 + 38\cdot 53^{2} + 19\cdot 53^{3} + 9\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 40 a + 5 + \left(5 a + 34\right)\cdot 53 + \left(3 a + 52\right)\cdot 53^{2} + \left(49 a + 16\right)\cdot 53^{3} + \left(4 a + 28\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 12 + 37\cdot 53 + 36\cdot 53^{2} + 24\cdot 53^{3} + 13\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 13 a + 6 + \left(47 a + 17\right)\cdot 53 + \left(49 a + 6\right)\cdot 53^{2} + \left(3 a + 51\right)\cdot 53^{3} + \left(48 a + 51\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 22 a + 35 + \left(34 a + 30\right)\cdot 53 + \left(28 a + 51\right)\cdot 53^{2} + \left(36 a + 43\right)\cdot 53^{3} + \left(50 a + 50\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 31 a + 17 + \left(18 a + 40\right)\cdot 53 + \left(24 a + 25\right)\cdot 53^{2} + \left(16 a + 2\right)\cdot 53^{3} + \left(2 a + 5\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
$c1$
$1$ $1$ $()$ $5$
$45$ $2$ $(1,2)(3,4)$ $1$
$40$ $3$ $(1,2,3)(4,5,6)$ $-1$
$40$ $3$ $(1,2,3)$ $2$
$90$ $4$ $(1,2,3,4)(5,6)$ $-1$
$72$ $5$ $(1,2,3,4,5)$ $0$
$72$ $5$ $(1,3,4,5,2)$ $0$
The blue line marks the conjugacy class containing complex conjugation.