Properties

Label 1.20.4t1.a.b
Dimension $1$
Group $C_4$
Conductor $20$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_4$
Conductor: \(20\)\(\medspace = 2^{2} \cdot 5 \)
Artin field: \(\Q(\zeta_{20})^+\)
Galois orbit size: $2$
Smallest permutation container: $C_4$
Parity: even
Dirichlet character: \(\chi_{20}(3,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{4} - 5 x^{2} + 5\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 19 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 6 + 6\cdot 19 + 18\cdot 19^{2} + 5\cdot 19^{3} + 12\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 8 + 13\cdot 19 + 6\cdot 19^{2} + 14\cdot 19^{3} + 11\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 11 + 5\cdot 19 + 12\cdot 19^{2} + 4\cdot 19^{3} + 7\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 13 + 12\cdot 19 + 13\cdot 19^{3} + 6\cdot 19^{4} +O(19^{5})\)  Toggle raw display

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

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

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

The blue line marks the conjugacy class containing complex conjugation.