Properties

Label 1.31.5t1.1c1
Dimension 1
Group $C_5$
Conductor $ 31 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_5$
Conductor:$31 $
Artin number field: Splitting field of $f= x^{5} - x^{4} - 12 x^{3} + 21 x^{2} + x - 5 $ over $\Q$
Size of Galois orbit: 4
Smallest containing permutation representation: $C_5$
Parity: Even
Corresponding Dirichlet character: \(\chi_{31}(8,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 37 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 10 + 20\cdot 37 + 28\cdot 37^{2} + 15\cdot 37^{3} + 5\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 18 + 26\cdot 37 + 8\cdot 37^{2} + 19\cdot 37^{3} + 20\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 20 + 8\cdot 37 + 17\cdot 37^{2} + 24\cdot 37^{3} + 4\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 31 + 20\cdot 37 + 10\cdot 37^{2} + 27\cdot 37^{3} + 10\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 33 + 34\cdot 37 + 8\cdot 37^{2} + 24\cdot 37^{3} + 32\cdot 37^{4} +O\left(37^{ 5 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 5 }$ Character value
$1$$1$$()$$1$
$1$$5$$(1,3,5,4,2)$$\zeta_{5}$
$1$$5$$(1,5,2,3,4)$$\zeta_{5}^{2}$
$1$$5$$(1,4,3,2,5)$$\zeta_{5}^{3}$
$1$$5$$(1,2,4,5,3)$$-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$
The blue line marks the conjugacy class containing complex conjugation.