Properties

Label 2.544.8t17.a
Dimension $2$
Group $C_4\wr C_2$
Conductor $544$
Indicator $0$

Related objects

Downloads

Learn more

Basic invariants

Dimension:$2$
Group:$C_4\wr C_2$
Conductor:\(544\)\(\medspace = 2^{5} \cdot 17 \)
Artin number field: Galois closure of 8.0.321978368.6
Galois orbit size: $2$
Smallest permutation container: $C_4\wr C_2$
Parity: odd
Projective image: $D_4$
Projective field: Galois closure of 4.2.39304.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 577 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ \( 62 + 295\cdot 577 + 138\cdot 577^{2} + 28\cdot 577^{3} + 561\cdot 577^{4} +O(577^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 284 + 218\cdot 577 + 538\cdot 577^{2} + 155\cdot 577^{3} + 10\cdot 577^{4} +O(577^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 291 + 463\cdot 577 + 94\cdot 577^{2} + 385\cdot 577^{3} + 313\cdot 577^{4} +O(577^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 375 + 138\cdot 577 + 285\cdot 577^{2} + 568\cdot 577^{3} + 39\cdot 577^{4} +O(577^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 433 + 520\cdot 577 + 575\cdot 577^{2} + 548\cdot 577^{3} + 502\cdot 577^{4} +O(577^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 438 + 559\cdot 577 + 329\cdot 577^{2} + 261\cdot 577^{3} + 347\cdot 577^{4} +O(577^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 481 + 487\cdot 577 + 359\cdot 577^{2} + 350\cdot 577^{3} + 176\cdot 577^{4} +O(577^{5})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 525 + 200\cdot 577 + 562\cdot 577^{2} + 8\cdot 577^{3} + 356\cdot 577^{4} +O(577^{5})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

Cycle notation
$(1,8)(2,6)(3,7)(4,5)$
$(1,5,8,4)(2,3,6,7)$
$(2,6)(3,7)$
$(1,3,8,7)(2,5,6,4)$
$(2,7,6,3)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character values
$c1$ $c2$
$1$ $1$ $()$ $2$ $2$
$1$ $2$ $(1,8)(2,6)(3,7)(4,5)$ $-2$ $-2$
$2$ $2$ $(2,6)(3,7)$ $0$ $0$
$4$ $2$ $(1,6)(2,8)(3,4)(5,7)$ $0$ $0$
$1$ $4$ $(1,5,8,4)(2,3,6,7)$ $-2 \zeta_{4}$ $2 \zeta_{4}$
$1$ $4$ $(1,4,8,5)(2,7,6,3)$ $2 \zeta_{4}$ $-2 \zeta_{4}$
$2$ $4$ $(2,7,6,3)$ $\zeta_{4} + 1$ $-\zeta_{4} + 1$
$2$ $4$ $(2,3,6,7)$ $-\zeta_{4} + 1$ $\zeta_{4} + 1$
$2$ $4$ $(1,8)(2,3,6,7)(4,5)$ $-\zeta_{4} - 1$ $\zeta_{4} - 1$
$2$ $4$ $(1,8)(2,7,6,3)(4,5)$ $\zeta_{4} - 1$ $-\zeta_{4} - 1$
$2$ $4$ $(1,5,8,4)(2,7,6,3)$ $0$ $0$
$4$ $4$ $(1,3,8,7)(2,5,6,4)$ $0$ $0$
$4$ $8$ $(1,2,5,3,8,6,4,7)$ $0$ $0$
$4$ $8$ $(1,3,4,2,8,7,5,6)$ $0$ $0$
The blue line marks the conjugacy class containing complex conjugation.