Properties

Label 3.2e4_83e2.6t8.2c1
Dimension 3
Group $S_4$
Conductor $ 2^{4} \cdot 83^{2}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$3$
Group:$S_4$
Conductor:$110224= 2^{4} \cdot 83^{2} $
Artin number field: Splitting field of $f= x^{4} - 3 x^{2} - 2 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_4$
Parity: Even
Determinant: 1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 227 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 66 + 4\cdot 227 + 212\cdot 227^{2} + 44\cdot 227^{3} + 20\cdot 227^{4} +O\left(227^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 82 + 109\cdot 227 + 218\cdot 227^{2} + 176\cdot 227^{3} + 97\cdot 227^{4} +O\left(227^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 90 + 59\cdot 227 + 215\cdot 227^{2} + 190\cdot 227^{3} + 130\cdot 227^{4} +O\left(227^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 216 + 53\cdot 227 + 35\cdot 227^{2} + 41\cdot 227^{3} + 205\cdot 227^{4} +O\left(227^{ 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.