Properties

Label 1.109.4t1.1c1
Dimension 1
Group $C_4$
Conductor $ 109 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_4$
Conductor:$109 $
Artin number field: Splitting field of $f= x^{4} - x^{3} + 14 x^{2} + 34 x + 393 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_4$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{109}(76,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 73 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 9 + 31\cdot 73 + 15\cdot 73^{2} + 26\cdot 73^{3} + 42\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 13 + 5\cdot 73 + 54\cdot 73^{2} + 54\cdot 73^{3} + 67\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 21 + 34\cdot 73 + 23\cdot 73^{2} + 62\cdot 73^{3} + 64\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 31 + 2\cdot 73 + 53\cdot 73^{2} + 2\cdot 73^{3} + 44\cdot 73^{4} +O\left(73^{ 5 }\right)$

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

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

Character values on conjugacy classes

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