Properties

Label 5.14250625.6t12.a.a
Dimension $5$
Group $A_5$
Conductor $14250625$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $5$
Group: $A_5$
Conductor: \(14250625\)\(\medspace = 5^{4} \cdot 151^{2}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 5.1.570025.1
Galois orbit size: $1$
Smallest permutation container: $\PSL(2,5)$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $A_5$
Projective stem field: 5.1.570025.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: \(x^{2} + 7 x + 2\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 4 a + 4 + 8\cdot 11 + \left(7 a + 4\right)\cdot 11^{2} + \left(10 a + 7\right)\cdot 11^{3} + \left(6 a + 5\right)\cdot 11^{4} +O(11^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 7 a + 9 + \left(10 a + 5\right)\cdot 11 + \left(3 a + 10\right)\cdot 11^{2} + 9\cdot 11^{3} + 4 a\cdot 11^{4} +O(11^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( a + 5 + \left(3 a + 9\right)\cdot 11 + \left(7 a + 2\right)\cdot 11^{2} + \left(10 a + 7\right)\cdot 11^{3} + \left(6 a + 10\right)\cdot 11^{4} +O(11^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 6 + 10\cdot 11 + 7\cdot 11^{2} + 9\cdot 11^{3} + 9\cdot 11^{4} +O(11^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 10 a + 9 + \left(7 a + 9\right)\cdot 11 + \left(3 a + 6\right)\cdot 11^{2} + 9\cdot 11^{3} + \left(4 a + 5\right)\cdot 11^{4} +O(11^{5})\)  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$$1$$()$$5$
$15$$2$$(1,2)(3,4)$$1$
$20$$3$$(1,2,3)$$-1$
$12$$5$$(1,2,3,4,5)$$0$
$12$$5$$(1,3,4,5,2)$$0$

The blue line marks the conjugacy class containing complex conjugation.