Properties

Label 3.1264.4t5.a
Dimension $3$
Group $S_4$
Conductor $1264$
Indicator $1$

Related objects

Downloads

Learn more

Basic invariants

Dimension:$3$
Group:$S_4$
Conductor:\(1264\)\(\medspace = 2^{4} \cdot 79 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 4.0.1264.1
Galois orbit size: $1$
Smallest permutation container: $S_4$
Parity: even
Projective image: $S_4$
Projective field: Galois closure of 4.0.1264.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 307 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ \( 49 + 112\cdot 307 + 243\cdot 307^{2} + 73\cdot 307^{3} + 7\cdot 307^{4} +O(307^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 53 + 155\cdot 307 + 297\cdot 307^{2} + 221\cdot 307^{3} + 73\cdot 307^{4} +O(307^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 87 + 171\cdot 307 + 221\cdot 307^{2} + 125\cdot 307^{3} + 119\cdot 307^{4} +O(307^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 118 + 175\cdot 307 + 158\cdot 307^{2} + 192\cdot 307^{3} + 106\cdot 307^{4} +O(307^{5})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 4 }$

Cycle notation
$(1,2,3,4)$
$(1,2)$

Character values on conjugacy classes

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