Properties

Label 1.65.4t1.d.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 number field: Galois closure of 4.4.274625.1
Galois orbit size: $2$
Smallest permutation container: $C_4$
Parity: even
Dirichlet character: \(\chi_{65}(8,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$$ x^{4} - x^{3} - 24 x^{2} + 4 x + 16 $.

The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ $ 12 + 42\cdot 61 + 60\cdot 61^{2} + 58\cdot 61^{3} + 30\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 20 + 11\cdot 61 + 33\cdot 61^{2} + 35\cdot 61^{3} + 11\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 45 + 61 + 11\cdot 61^{2} + 34\cdot 61^{3} + 38\cdot 61^{4} +O\left(61^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 46 + 5\cdot 61 + 17\cdot 61^{2} + 54\cdot 61^{3} + 40\cdot 61^{4} +O\left(61^{ 5 }\right)$

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

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

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$$4$$(1,3,2,4)$$\zeta_{4}$
$1$$4$$(1,4,2,3)$$-\zeta_{4}$

The blue line marks the conjugacy class containing complex conjugation.