Properties

Label 1.120.4t1.b.b
Dimension $1$
Group $C_4$
Conductor $120$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_4$
Conductor: \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \)
Artin field: Galois closure of 4.0.72000.2
Galois orbit size: $2$
Smallest permutation container: $C_4$
Parity: odd
Dirichlet character: \(\chi_{120}(107,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

$f(x)$$=$ \( x^{4} + 30x^{2} + 180 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 3 + 35\cdot 59 + 25\cdot 59^{2} + 13\cdot 59^{3} + 7\cdot 59^{4} +O(59^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 16 + 32\cdot 59 + 35\cdot 59^{2} + 48\cdot 59^{3} + 31\cdot 59^{4} +O(59^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 43 + 26\cdot 59 + 23\cdot 59^{2} + 10\cdot 59^{3} + 27\cdot 59^{4} +O(59^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 56 + 23\cdot 59 + 33\cdot 59^{2} + 45\cdot 59^{3} + 51\cdot 59^{4} +O(59^{5})\) Copy content 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.