Properties

Label 1.40.4t1.b.b
Dimension $1$
Group $C_4$
Conductor $40$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_4$
Conductor: \(40\)\(\medspace = 2^{3} \cdot 5 \)
Artin field: 4.0.8000.2
Galois orbit size: $2$
Smallest permutation container: $C_4$
Parity: odd
Dirichlet character: \(\chi_{40}(37,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{4} + 10 x^{2} + 20\)  Toggle raw display.

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

Roots:
$r_{ 1 }$ $=$ \( 2 + 19 + 12\cdot 19^{3} + 2\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 9 + 9\cdot 19 + 15\cdot 19^{2} + 14\cdot 19^{3} + 8\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 10 + 9\cdot 19 + 3\cdot 19^{2} + 4\cdot 19^{3} + 10\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 17 + 17\cdot 19 + 18\cdot 19^{2} + 6\cdot 19^{3} + 16\cdot 19^{4} +O(19^{5})\)  Toggle raw display

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.