Properties

Label 1.240.4t1.d.b
Dimension $1$
Group $C_4$
Conductor $240$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_4$
Conductor: \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \)
Artin field: 4.0.2304000.1
Galois orbit size: $2$
Smallest permutation container: $C_4$
Parity: odd
Dirichlet character: \(\chi_{240}(83,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{4} + 60 x^{2} + 90\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 41 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 14 + 6\cdot 41 + 10\cdot 41^{2} + 33\cdot 41^{3} + 18\cdot 41^{4} +O(41^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 20 + 20\cdot 41 + 2\cdot 41^{2} + 37\cdot 41^{3} + 6\cdot 41^{4} +O(41^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 21 + 20\cdot 41 + 38\cdot 41^{2} + 3\cdot 41^{3} + 34\cdot 41^{4} +O(41^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 27 + 34\cdot 41 + 30\cdot 41^{2} + 7\cdot 41^{3} + 22\cdot 41^{4} +O(41^{5})\)  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.