Properties

Label 1.165.4t1.a.a
Dimension $1$
Group $C_4$
Conductor $165$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_4$
Conductor: \(165\)\(\medspace = 3 \cdot 5 \cdot 11 \)
Artin field: Galois closure of 4.0.136125.2
Galois orbit size: $2$
Smallest permutation container: $C_4$
Parity: odd
Dirichlet character: \(\chi_{165}(32,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

$f(x)$$=$ \( x^{4} - x^{3} + 41x^{2} - 41x + 361 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 19 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 13\cdot 19^{2} + 16\cdot 19^{3} + 5\cdot 19^{4} +O(19^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 1 + 14\cdot 19^{2} + 16\cdot 19^{3} + 15\cdot 19^{4} +O(19^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 4 + 2\cdot 19 + 15\cdot 19^{2} + 16\cdot 19^{3} + 10\cdot 19^{4} +O(19^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 15 + 16\cdot 19 + 14\cdot 19^{2} + 6\cdot 19^{3} + 5\cdot 19^{4} +O(19^{5})\) Copy content Toggle raw display

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

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

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$$4$$(1,2,4,3)$$\zeta_{4}$
$1$$4$$(1,3,4,2)$$-\zeta_{4}$

The blue line marks the conjugacy class containing complex conjugation.