Properties

Label 1.71.5t1.a.d
Dimension $1$
Group $C_5$
Conductor $71$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_5$
Conductor: \(71\)
Artin field: Galois closure of 5.5.25411681.1
Galois orbit size: $4$
Smallest permutation container: $C_5$
Parity: even
Dirichlet character: \(\chi_{71}(25,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

$f(x)$$=$ \( x^{5} - x^{4} - 28x^{3} - 37x^{2} + 25x - 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 37 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 4 + 4\cdot 37 + 30\cdot 37^{2} + 6\cdot 37^{3} + 5\cdot 37^{4} +O(37^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 22 + 36\cdot 37 + 12\cdot 37^{2} + 33\cdot 37^{3} + 22\cdot 37^{4} +O(37^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 23 + 18\cdot 37 + 10\cdot 37^{2} + 11\cdot 37^{3} + 13\cdot 37^{4} +O(37^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 27 + 34\cdot 37 + 32\cdot 37^{2} + 2\cdot 37^{3} + 31\cdot 37^{4} +O(37^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 36 + 16\cdot 37 + 24\cdot 37^{2} + 19\cdot 37^{3} + 37^{4} +O(37^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.