Properties

Label 1.260.4t1.b.b
Dimension $1$
Group $C_4$
Conductor $260$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_4$
Conductor: \(260\)\(\medspace = 2^{2} \cdot 5 \cdot 13 \)
Artin field: Galois closure of 4.0.4394000.1
Galois orbit size: $2$
Smallest permutation container: $C_4$
Parity: odd
Dirichlet character: \(\chi_{260}(83,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

$f(x)$$=$ \( x^{4} + 65x^{2} + 260 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 37 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 7 + 7\cdot 37 + 13\cdot 37^{2} + 4\cdot 37^{3} + 17\cdot 37^{4} +O(37^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 16 + 29\cdot 37 + 35\cdot 37^{2} + 29\cdot 37^{3} + 33\cdot 37^{4} +O(37^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 21 + 7\cdot 37 + 37^{2} + 7\cdot 37^{3} + 3\cdot 37^{4} +O(37^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 30 + 29\cdot 37 + 23\cdot 37^{2} + 32\cdot 37^{3} + 19\cdot 37^{4} +O(37^{5})\) 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.