Properties

Label 4.7_19_29.5t5.1
Dimension 4
Group $S_5$
Conductor $ 7 \cdot 19 \cdot 29 $
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$4$
Group:$S_5$
Conductor:$3857= 7 \cdot 19 \cdot 29 $
Artin number field: Splitting field of $f= x^{5} - x^{4} + x^{3} - x^{2} + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_5$
Parity: Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: $ x^{2} + 60 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 10 a + 57 + \left(37 a + 10\right)\cdot 61 + \left(9 a + 57\right)\cdot 61^{2} + 33\cdot 61^{3} + \left(4 a + 4\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 51 a + 6 + \left(23 a + 38\right)\cdot 61 + \left(51 a + 29\right)\cdot 61^{2} + \left(60 a + 24\right)\cdot 61^{3} + \left(56 a + 8\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 31 a + 11 + \left(53 a + 28\right)\cdot 61 + \left(30 a + 43\right)\cdot 61^{2} + \left(8 a + 37\right)\cdot 61^{3} + \left(21 a + 3\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 7 + 55\cdot 61 + 31\cdot 61^{2} + 10\cdot 61^{3} + 28\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 30 a + 42 + \left(7 a + 50\right)\cdot 61 + \left(30 a + 20\right)\cdot 61^{2} + \left(52 a + 15\right)\cdot 61^{3} + \left(39 a + 16\right)\cdot 61^{4} +O\left(61^{ 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.