Properties

Label 1.2e4_5.4t1.6
Dimension 1
Group $C_4$
Conductor $ 2^{4} \cdot 5 $
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_4$
Conductor:$80= 2^{4} \cdot 5 $
Artin number field: Splitting field of $f= x^{4} + 20 x^{2} + 90 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_4$
Parity: Odd

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 13 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 3 + 11\cdot 13 + 6\cdot 13^{2} + 7\cdot 13^{3} + 6\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 6 + 6\cdot 13 + 9\cdot 13^{2} + 7\cdot 13^{3} + 5\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 7 + 6\cdot 13 + 3\cdot 13^{2} + 5\cdot 13^{3} + 7\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 10 + 13 + 6\cdot 13^{2} + 5\cdot 13^{3} + 6\cdot 13^{4} +O\left(13^{ 5 }\right)$

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 values
$c1$ $c2$
$1$ $1$ $()$ $1$ $1$
$1$ $2$ $(1,4)(2,3)$ $-1$ $-1$
$1$ $4$ $(1,3,4,2)$ $\zeta_{4}$ $-\zeta_{4}$
$1$ $4$ $(1,2,4,3)$ $-\zeta_{4}$ $\zeta_{4}$
The blue line marks the conjugacy class containing complex conjugation.