Properties

Label 1.5.4t1.a
Dimension $1$
Group $C_4$
Conductor $5$
Indicator $0$

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Basic invariants

Dimension:$1$
Group:$C_4$
Conductor:\(5\)
Artin number field: Galois closure of \(\Q(\zeta_{5})\)
Galois orbit size: $2$
Smallest permutation container: $C_4$
Parity: odd
Projective image: $C_1$
Projective field: \(\Q\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 11 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ \( 2 + 10\cdot 11 + 4\cdot 11^{2} + 9\cdot 11^{3} + 11^{4} +O(11^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 6 + 8\cdot 11 + 5\cdot 11^{2} + 9\cdot 11^{3} + 3\cdot 11^{4} +O(11^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 7 + 3\cdot 11 + 11^{2} + 5\cdot 11^{3} + 8\cdot 11^{4} +O(11^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 8 + 10\cdot 11 + 9\cdot 11^{2} + 8\cdot 11^{3} + 7\cdot 11^{4} +O(11^{5})\)  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 values
$c1$ $c2$
$1$ $1$ $()$ $1$ $1$
$1$ $2$ $(1,2)(3,4)$ $-1$ $-1$
$1$ $4$ $(1,4,2,3)$ $\zeta_{4}$ $-\zeta_{4}$
$1$ $4$ $(1,3,2,4)$ $-\zeta_{4}$ $\zeta_{4}$
The blue line marks the conjugacy class containing complex conjugation.