Properties

Label 1.304.4t1.b
Dimension $1$
Group $C_4$
Conductor $304$
Indicator $0$

Related objects

Downloads

Learn more

Basic invariants

Dimension:$1$
Group:$C_4$
Conductor:\(304\)\(\medspace = 2^{4} \cdot 19 \)
Artin number field: Galois closure of 4.0.739328.2
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 6.
Roots:
$r_{ 1 }$ $=$ \( 1 + 7\cdot 17 + 4\cdot 17^{2} + 5\cdot 17^{3} + 13\cdot 17^{4} + 6\cdot 17^{5} +O(17^{6})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 5 + 15\cdot 17 + 15\cdot 17^{2} + 7\cdot 17^{3} + 8\cdot 17^{4} + 11\cdot 17^{5} +O(17^{6})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 12 + 17 + 17^{2} + 9\cdot 17^{3} + 8\cdot 17^{4} + 5\cdot 17^{5} +O(17^{6})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 16 + 9\cdot 17 + 12\cdot 17^{2} + 11\cdot 17^{3} + 3\cdot 17^{4} + 10\cdot 17^{5} +O(17^{6})\) Copy content Toggle raw display

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.