Properties

Label 1.17_59.4t1.1c2
Dimension 1
Group $C_4$
Conductor $ 17 \cdot 59 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_4$
Conductor:$1003= 17 \cdot 59 $
Artin number field: Splitting field of $f= x^{4} - x^{3} + 249 x^{2} + x + 15046 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_4$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{1003}(353,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 83 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 18 + 49\cdot 83 + 18\cdot 83^{2} + 81\cdot 83^{3} + 54\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 19 + 19\cdot 83 + 52\cdot 83^{2} + 3\cdot 83^{3} + 37\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 56 + 52\cdot 83 + 29\cdot 83^{2} + 48\cdot 83^{3} + 62\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 74 + 44\cdot 83 + 65\cdot 83^{2} + 32\cdot 83^{3} + 11\cdot 83^{4} +O\left(83^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 4 }$

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

Character values on conjugacy classes

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