Properties

Label 1.25.5t1.a.a
Dimension $1$
Group $C_5$
Conductor $25$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_5$
Conductor: \(25\)\(\medspace = 5^{2}\)
Artin field: 5.5.390625.1
Galois orbit size: $4$
Smallest permutation container: $C_5$
Parity: even
Dirichlet character: \(\chi_{25}(11,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{5} - 10 x^{3} - 5 x^{2} + 10 x - 1\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 3 + 3\cdot 43 + 24\cdot 43^{2} + 5\cdot 43^{3} + 7\cdot 43^{4} +O(43^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 7 + 13\cdot 43 + 19\cdot 43^{2} + 5\cdot 43^{3} + 21\cdot 43^{4} +O(43^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 19 + 14\cdot 43 + 26\cdot 43^{2} + 32\cdot 43^{3} + 5\cdot 43^{4} +O(43^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 24 + 12\cdot 43 + 2\cdot 43^{2} + 38\cdot 43^{3} + 26\cdot 43^{4} +O(43^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 33 + 42\cdot 43 + 13\cdot 43^{2} + 4\cdot 43^{3} + 25\cdot 43^{4} +O(43^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.