Properties

Label 1.65.4t1.b
Dimension $1$
Group $C_4$
Conductor $65$
Indicator $0$

Related objects

Downloads

Learn more

Basic invariants

Dimension:$1$
Group:$C_4$
Conductor:\(65\)\(\medspace = 5 \cdot 13 \)
Artin number field: Galois closure of 4.0.54925.1
Galois orbit size: $2$
Smallest permutation container: $C_4$
Parity: odd
Projective image: $C_1$
Projective field: Galois closure of \(\Q\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 17 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ \( 6 + 16\cdot 17 + 6\cdot 17^{2} + 2\cdot 17^{3} + 2\cdot 17^{4} +O(17^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 7 + 2\cdot 17 + 16\cdot 17^{2} + 5\cdot 17^{3} + 9\cdot 17^{4} +O(17^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 8 + 16\cdot 17 + 16\cdot 17^{3} + 12\cdot 17^{4} +O(17^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 14 + 15\cdot 17 + 9\cdot 17^{2} + 9\cdot 17^{3} + 9\cdot 17^{4} +O(17^{5})\) Copy content Toggle raw display

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