Properties

Label 5.2e14_5e8.6t15.1
Dimension 5
Group $A_6$
Conductor $ 2^{14} \cdot 5^{8}$
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$5$
Group:$A_6$
Conductor:$6400000000= 2^{14} \cdot 5^{8} $
Artin number field: Splitting field of $f= x^{6} - 2 x^{5} + 15 x^{4} + 50 x^{2} - 4 x - 82 $ 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 $\Q_{ 23 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $ x^{2} + 21 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 3 a + 16 + \left(19 a + 17\right)\cdot 23 + \left(a + 19\right)\cdot 23^{2} + \left(13 a + 5\right)\cdot 23^{3} + 12\cdot 23^{4} + \left(9 a + 1\right)\cdot 23^{5} + \left(11 a + 17\right)\cdot 23^{6} + \left(18 a + 18\right)\cdot 23^{7} +O\left(23^{ 8 }\right)$
$r_{ 2 }$ $=$ $ 11 + 16\cdot 23 + 15\cdot 23^{2} + 15\cdot 23^{3} + 3\cdot 23^{4} + 14\cdot 23^{5} + 19\cdot 23^{6} + 23^{7} +O\left(23^{ 8 }\right)$
$r_{ 3 }$ $=$ $ 7 a + 6 + \left(12 a + 8\right)\cdot 23 + \left(11 a + 7\right)\cdot 23^{2} + \left(6 a + 14\right)\cdot 23^{3} + \left(13 a + 7\right)\cdot 23^{4} + \left(5 a + 5\right)\cdot 23^{5} + 18 a\cdot 23^{6} + \left(19 a + 10\right)\cdot 23^{7} +O\left(23^{ 8 }\right)$
$r_{ 4 }$ $=$ $ 20 a + 22 + \left(3 a + 6\right)\cdot 23 + \left(21 a + 4\right)\cdot 23^{2} + \left(9 a + 7\right)\cdot 23^{3} + 22 a\cdot 23^{4} + \left(13 a + 19\right)\cdot 23^{5} + \left(11 a + 7\right)\cdot 23^{6} + \left(4 a + 21\right)\cdot 23^{7} +O\left(23^{ 8 }\right)$
$r_{ 5 }$ $=$ $ 19 + 16\cdot 23 + 3\cdot 23^{2} + 10\cdot 23^{3} + 17\cdot 23^{4} + 2\cdot 23^{5} + 16\cdot 23^{6} + 8\cdot 23^{7} +O\left(23^{ 8 }\right)$
$r_{ 6 }$ $=$ $ 16 a + 20 + \left(10 a + 2\right)\cdot 23 + \left(11 a + 18\right)\cdot 23^{2} + \left(16 a + 15\right)\cdot 23^{3} + \left(9 a + 4\right)\cdot 23^{4} + \left(17 a + 3\right)\cdot 23^{5} + \left(4 a + 8\right)\cdot 23^{6} + \left(3 a + 8\right)\cdot 23^{7} +O\left(23^{ 8 }\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.