Properties

Label 4.2617.5t5.b
Dimension 4
Group $S_5$
Conductor $ 2617 $
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$S_5$
Conductor:$2617 $
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: Even
Projective image: $S_5$
Projective field: Galois closure of 5.1.2617.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{2} + 24 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 11 a + 13 + \left(16 a + 17\right)\cdot 29 + \left(12 a + 18\right)\cdot 29^{2} + \left(21 a + 28\right)\cdot 29^{3} + \left(4 a + 19\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 18 a + 10 + \left(12 a + 1\right)\cdot 29 + \left(16 a + 7\right)\cdot 29^{2} + \left(7 a + 7\right)\cdot 29^{3} + \left(24 a + 22\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 20 a + 20 + \left(10 a + 21\right)\cdot 29 + \left(27 a + 26\right)\cdot 29^{2} + \left(10 a + 10\right)\cdot 29^{3} + \left(14 a + 27\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 9 a + 4 + \left(18 a + 26\right)\cdot 29 + \left(a + 7\right)\cdot 29^{2} + \left(18 a + 9\right)\cdot 29^{3} + \left(14 a + 1\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 11 + 20\cdot 29 + 26\cdot 29^{2} + 29^{3} + 16\cdot 29^{4} +O\left(29^{ 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.