Properties

Label 1.145.4t1.b.a
Dimension $1$
Group $C_4$
Conductor $145$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_4$
Conductor: \(145\)\(\medspace = 5 \cdot 29 \)
Artin field: 4.0.105125.2
Galois orbit size: $2$
Smallest permutation container: $C_4$
Parity: odd
Dirichlet character: \(\chi_{145}(57,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{4} - x^{3} + 36 x^{2} - 36 x + 281\)  Toggle raw display.

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

Roots:
$r_{ 1 }$ $=$ \( 2 + 4\cdot 19^{2} + 13\cdot 19^{3} + 12\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 3 + 2\cdot 19 + 6\cdot 19^{2} + 19^{3} + 14\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 5 + 2\cdot 19 + 11\cdot 19^{2} + 12\cdot 19^{3} + 4\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 10 + 14\cdot 19 + 16\cdot 19^{2} + 10\cdot 19^{3} + 6\cdot 19^{4} +O(19^{5})\)  Toggle raw display

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

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

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

The blue line marks the conjugacy class containing complex conjugation.