Properties

Label 1.143.4t1.a.b
Dimension $1$
Group $C_4$
Conductor $143$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_4$
Conductor: \(143\)\(\medspace = 11 \cdot 13 \)
Artin field: Galois closure of 4.4.265837.1
Galois orbit size: $2$
Smallest permutation container: $C_4$
Parity: even
Dirichlet character: \(\chi_{143}(21,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

$f(x)$$=$ \( x^{4} - x^{3} - 37x^{2} - 35x + 81 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 2 + 29\cdot 43 + 39\cdot 43^{2} + 35\cdot 43^{3} + 23\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 10 + 12\cdot 43 + 43^{2} + 31\cdot 43^{3} +O(43^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 13 + 16\cdot 43 + 11\cdot 43^{2} + 7\cdot 43^{3} + 3\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 19 + 28\cdot 43 + 33\cdot 43^{2} + 11\cdot 43^{3} + 15\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display

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

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

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.