Properties

Label 4.13_347.5t5.1
Dimension 4
Group $S_5$
Conductor $ 13 \cdot 347 $
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$4$
Group:$S_5$
Conductor:$4511= 13 \cdot 347 $
Artin number field: Splitting field of $f= x^{5} - x^{3} - 2 x^{2} + 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 $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{2} + 33 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 7 a + 16 + \left(5 a + 23\right)\cdot 37 + \left(20 a + 6\right)\cdot 37^{2} + \left(35 a + 36\right)\cdot 37^{3} + \left(22 a + 25\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 21 + 6\cdot 37 + 6\cdot 37^{2} + 5\cdot 37^{3} + 27\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 18 a + 16 + \left(6 a + 36\right)\cdot 37 + \left(10 a + 27\right)\cdot 37^{2} + \left(8 a + 36\right)\cdot 37^{3} + \left(11 a + 24\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 30 a + 7 + 31 a\cdot 37 + \left(16 a + 8\right)\cdot 37^{2} + \left(a + 10\right)\cdot 37^{3} + \left(14 a + 8\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 19 a + 14 + \left(30 a + 7\right)\cdot 37 + \left(26 a + 25\right)\cdot 37^{2} + \left(28 a + 22\right)\cdot 37^{3} + \left(25 a + 24\right)\cdot 37^{4} +O\left(37^{ 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.