Properties

Label 1.61.5t1.a.d
Dimension $1$
Group $C_5$
Conductor $61$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_5$
Conductor: \(61\)
Artin field: 5.5.13845841.1
Galois orbit size: $4$
Smallest permutation container: $C_5$
Parity: even
Dirichlet character: \(\chi_{61}(58,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{5} - x^{4} - 24 x^{3} + 17 x^{2} + 41 x + 13\)  Toggle raw display.

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

Roots:
$r_{ 1 }$ $=$ \( 4 + 8\cdot 11 + 8\cdot 11^{2} + 2\cdot 11^{3} + 3\cdot 11^{4} +O(11^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 5 + 10\cdot 11 + 8\cdot 11^{2} + 5\cdot 11^{3} + 6\cdot 11^{4} +O(11^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 6 + 6\cdot 11 + 7\cdot 11^{2} + 3\cdot 11^{3} +O(11^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 9 + 6\cdot 11 + 5\cdot 11^{2} + 6\cdot 11^{3} + 8\cdot 11^{4} +O(11^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 10 + 2\cdot 11^{2} + 3\cdot 11^{3} + 3\cdot 11^{4} +O(11^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.