Properties

Label 1.2e3_13.4t1.2
Dimension 1
Group $C_4$
Conductor $ 2^{3} \cdot 13 $
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_4$
Conductor:$104= 2^{3} \cdot 13 $
Artin number field: Splitting field of $f= x^{4} + 26 x^{2} + 52 $ 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 $\Q_{ 43 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 2 + 24\cdot 43 + 9\cdot 43^{2} + 3\cdot 43^{3} + 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 20 + 21\cdot 43 + 13\cdot 43^{3} + 8\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 23 + 21\cdot 43 + 42\cdot 43^{2} + 29\cdot 43^{3} + 34\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 41 + 18\cdot 43 + 33\cdot 43^{2} + 39\cdot 43^{3} + 41\cdot 43^{4} +O\left(43^{ 5 }\right)$

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