Properties

Label 1.240.4t1.c.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: 4.4.2304000.1
Galois orbit size: $2$
Smallest permutation container: $C_4$
Parity: even
Dirichlet character: \(\chi_{240}(197,\cdot)\)
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 $\Q_{ 31 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 1 + 27\cdot 31 + 16\cdot 31^{2} + 5\cdot 31^{3} + 2\cdot 31^{4} +O(31^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 11 + 20\cdot 31 + 24\cdot 31^{2} + 17\cdot 31^{3} + 24\cdot 31^{4} +O(31^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 20 + 10\cdot 31 + 6\cdot 31^{2} + 13\cdot 31^{3} + 6\cdot 31^{4} +O(31^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 30 + 3\cdot 31 + 14\cdot 31^{2} + 25\cdot 31^{3} + 28\cdot 31^{4} +O(31^{5})\)  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.