Properties

Label 3.149769.12t33.a.b
Dimension $3$
Group $A_5$
Conductor $149769$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $3$
Group: $A_5$
Conductor: \(149769\)\(\medspace = 3^{4} \cdot 43^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 5.1.149769.1
Galois orbit size: $2$
Smallest permutation container: $A_5$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $A_5$
Projective stem field: Galois closure of 5.1.149769.1

Defining polynomial

$f(x)$$=$ \( x^{5} - 2x^{4} + x^{3} - 5x^{2} + x - 2 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: \( x^{2} + 24x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 16 a + 19 + \left(22 a + 21\right)\cdot 29 + \left(14 a + 2\right)\cdot 29^{2} + \left(3 a + 1\right)\cdot 29^{3} + \left(6 a + 26\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 24 a + 5 + \left(14 a + 18\right)\cdot 29 + 21 a\cdot 29^{2} + \left(5 a + 21\right)\cdot 29^{3} + \left(12 a + 16\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 5 a + 9 + \left(14 a + 10\right)\cdot 29 + \left(7 a + 6\right)\cdot 29^{2} + \left(23 a + 28\right)\cdot 29^{3} + \left(16 a + 13\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 13 a + 12 + \left(6 a + 2\right)\cdot 29 + \left(14 a + 25\right)\cdot 29^{2} + \left(25 a + 3\right)\cdot 29^{3} + \left(22 a + 24\right)\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 15 + 5\cdot 29 + 23\cdot 29^{2} + 3\cdot 29^{3} + 6\cdot 29^{4} +O(29^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 5 }$ Character value
$1$$1$$()$$3$
$15$$2$$(1,2)(3,4)$$-1$
$20$$3$$(1,2,3)$$0$
$12$$5$$(1,2,3,4,5)$$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$
$12$$5$$(1,3,4,5,2)$$-\zeta_{5}^{3} - \zeta_{5}^{2}$

The blue line marks the conjugacy class containing complex conjugation.