Properties

Label 1.41.5t1.a.a
Dimension $1$
Group $C_5$
Conductor $41$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_5$
Conductor: \(41\)
Artin number field: Galois closure of 5.5.2825761.1
Galois orbit size: $4$
Smallest permutation container: $C_5$
Parity: even
Dirichlet character: \(\chi_{41}(16,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$$ x^{5} - x^{4} - 16 x^{3} - 5 x^{2} + 21 x + 9 $.

The roots of $f$ are computed in $\Q_{ 73 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ $ 41 + 65\cdot 73 + 54\cdot 73^{2} + 73^{3} + 5\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 53 + 28\cdot 73 + 29\cdot 73^{2} + 32\cdot 73^{3} + 6\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 64 + 69\cdot 73 + 57\cdot 73^{2} + 38\cdot 73^{3} + 42\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 67 + 26\cdot 73 + 36\cdot 73^{2} + 59\cdot 73^{3} + 47\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 68 + 27\cdot 73 + 40\cdot 73^{2} + 13\cdot 73^{3} + 44\cdot 73^{4} +O\left(73^{ 5 }\right)$

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.