Properties

Label 1.3_5_67.4t1.1c2
Dimension 1
Group $C_4$
Conductor $ 3 \cdot 5 \cdot 67 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_4$
Conductor:$1005= 3 \cdot 5 \cdot 67 $
Artin number field: Splitting field of $f= x^{4} - x^{3} + 251 x^{2} - 251 x + 12751 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_4$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{1005}(803,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 29 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 10 + 8\cdot 29 + 22\cdot 29^{2} + 10\cdot 29^{3} + 11\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 13 + 29 + 26\cdot 29^{2} + 4\cdot 29^{3} + 23\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 14 + 28\cdot 29 + 29^{2} + 19\cdot 29^{3} + 2\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 22 + 19\cdot 29 + 7\cdot 29^{2} + 23\cdot 29^{3} + 20\cdot 29^{4} +O\left(29^{ 5 }\right)$

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

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

Character values on conjugacy classes

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