Properties

Label 1.105.4t1.a.b
Dimension $1$
Group $C_4$
Conductor $105$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_4$
Conductor: \(105\)\(\medspace = 3 \cdot 5 \cdot 7 \)
Artin field: 4.0.55125.1
Galois orbit size: $2$
Smallest permutation container: $C_4$
Parity: odd
Dirichlet character: \(\chi_{105}(62,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{4} - x^{3} + 26 x^{2} - 26 x + 151\)  Toggle raw display.

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

Roots:
$r_{ 1 }$ $=$ \( 3 + 9\cdot 19^{2} + 7\cdot 19^{3} + 9\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 11 + 14\cdot 19 + 18\cdot 19^{3} + 12\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 12 + 16\cdot 19 + 18\cdot 19^{2} + 15\cdot 19^{3} + 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 13 + 6\cdot 19 + 9\cdot 19^{2} + 15\cdot 19^{3} + 13\cdot 19^{4} +O(19^{5})\)  Toggle raw display

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

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

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

The blue line marks the conjugacy class containing complex conjugation.