Properties

Label 3.2e2_397.4t5.1c1
Dimension 3
Group $S_4$
Conductor $ 2^{2} \cdot 397 $
Root number 1
Frobenius-Schur indicator 1

Related objects

Learn more about

Basic invariants

Dimension:$3$
Group:$S_4$
Conductor:$1588= 2^{2} \cdot 397 $
Artin number field: Splitting field of $f= x^{4} - x^{3} - 3 x^{2} + 2 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_4$
Parity: Odd
Determinant: 1.2e2_397.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 211 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 47 + 116\cdot 211 + 174\cdot 211^{2} + 132\cdot 211^{3} + 108\cdot 211^{4} +O\left(211^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 116 + 76\cdot 211 + 193\cdot 211^{2} + 57\cdot 211^{3} + 100\cdot 211^{4} +O\left(211^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 117 + 168\cdot 211 + 14\cdot 211^{2} + 66\cdot 211^{3} + 2\cdot 211^{4} +O\left(211^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 143 + 60\cdot 211 + 39\cdot 211^{2} + 165\cdot 211^{3} + 210\cdot 211^{4} +O\left(211^{ 5 }\right)$

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 value
$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.