Properties

Label 4.18583e3.10t12.1
Dimension 4
Group $S_5$
Conductor $ 18583^{3}$
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$4$
Group:$S_5$
Conductor:$6417228161287= 18583^{3} $
Artin number field: Splitting field of $f= x^{5} - x^{4} - x^{3} - 2 x^{2} + x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_5$
Parity: Odd

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$: $ x^{2} + 6 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 2 a + 6 + \left(2 a + 1\right)\cdot 7 + \left(2 a + 6\right)\cdot 7^{2} + 4 a\cdot 7^{3} + 5\cdot 7^{4} +O\left(7^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 2 a + 4 + 4 a\cdot 7 + \left(4 a + 3\right)\cdot 7^{2} + \left(3 a + 1\right)\cdot 7^{3} + \left(4 a + 1\right)\cdot 7^{4} +O\left(7^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 5 a + 1 + \left(4 a + 2\right)\cdot 7 + \left(4 a + 6\right)\cdot 7^{2} + \left(2 a + 2\right)\cdot 7^{3} + \left(6 a + 1\right)\cdot 7^{4} +O\left(7^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 5 + 6\cdot 7 + 7^{2} + 7^{3} + 4\cdot 7^{4} +O\left(7^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 5 a + 6 + \left(2 a + 2\right)\cdot 7 + \left(2 a + 3\right)\cdot 7^{2} + 3 a\cdot 7^{3} + \left(2 a + 2\right)\cdot 7^{4} +O\left(7^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 5 }$

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 5 }$ Character values
$c1$
$1$ $1$ $()$ $4$
$10$ $2$ $(1,2)$ $-2$
$15$ $2$ $(1,2)(3,4)$ $0$
$20$ $3$ $(1,2,3)$ $1$
$30$ $4$ $(1,2,3,4)$ $0$
$24$ $5$ $(1,2,3,4,5)$ $-1$
$20$ $6$ $(1,2,3)(4,5)$ $1$
The blue line marks the conjugacy class containing complex conjugation.