Properties

Label 1.65.4t1.a.a
Dimension $1$
Group $C_4$
Conductor $65$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_4$
Conductor: \(65\)\(\medspace = 5 \cdot 13 \)
Artin field: Galois closure of 4.0.21125.1
Galois orbit size: $2$
Smallest permutation container: $C_4$
Parity: odd
Dirichlet character: \(\chi_{65}(38,\cdot)\)
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Defining polynomial

$f(x)$$=$ \( x^{4} - x^{3} + 16x^{2} - 16x + 61 \) Copy content Toggle raw display .

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

Roots:
$r_{ 1 }$ $=$ \( 12 + 43\cdot 59 + 3\cdot 59^{2} + 23\cdot 59^{3} + 9\cdot 59^{4} +O(59^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 14 + 10\cdot 59 + 22\cdot 59^{2} + 22\cdot 59^{3} + 27\cdot 59^{4} +O(59^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 43 + 10\cdot 59 + 27\cdot 59^{2} + 59^{3} + 17\cdot 59^{4} +O(59^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 50 + 53\cdot 59 + 5\cdot 59^{2} + 12\cdot 59^{3} + 5\cdot 59^{4} +O(59^{5})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 4 }$

Cycle notation
$(1,4,2,3)$
$(1,2)(3,4)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 4 }$ Character value
$1$$1$$()$$1$
$1$$2$$(1,2)(3,4)$$-1$
$1$$4$$(1,4,2,3)$$\zeta_{4}$
$1$$4$$(1,3,2,4)$$-\zeta_{4}$

The blue line marks the conjugacy class containing complex conjugation.