Properties

Label 4.2869.5t5.a.a
Dimension 4
Group $S_5$
Conductor $ 19 \cdot 151 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$S_5$
Conductor:$2869= 19 \cdot 151 $
Artin number field: Splitting field of 5.1.2869.1 defined by $f= x^{5} - x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_5$
Parity: Even
Determinant: 1.2869.2t1.a.a
Projective image: $S_5$
Projective field: Galois closure of 5.1.2869.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: $ x^{2} + 63 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 37 a + 11 + \left(52 a + 51\right)\cdot 67 + \left(a + 63\right)\cdot 67^{2} + \left(31 a + 51\right)\cdot 67^{3} + 43\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 34 a + 41 + \left(44 a + 28\right)\cdot 67 + \left(12 a + 3\right)\cdot 67^{2} + \left(18 a + 16\right)\cdot 67^{3} + \left(36 a + 60\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 14 + 59\cdot 67 + 38\cdot 67^{2} + 16\cdot 67^{3} + 29\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 30 a + 25 + \left(14 a + 23\right)\cdot 67 + \left(65 a + 18\right)\cdot 67^{2} + \left(35 a + 40\right)\cdot 67^{3} + \left(66 a + 14\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 33 a + 43 + \left(22 a + 38\right)\cdot 67 + \left(54 a + 9\right)\cdot 67^{2} + \left(48 a + 9\right)\cdot 67^{3} + \left(30 a + 53\right)\cdot 67^{4} +O\left(67^{ 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 value
$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.