Properties

Label 3.2e3_127.4t5.1c1
Dimension 3
Group $S_4$
Conductor $ 2^{3} \cdot 127 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$3$
Group:$S_4$
Conductor:$1016= 2^{3} \cdot 127 $
Artin number field: Splitting field of $f= x^{4} - x^{3} + x^{2} - 2 x + 2 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_4$
Parity: Even
Determinant: 1.2e3_127.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 173 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 48 + 75\cdot 173 + 91\cdot 173^{2} + 119\cdot 173^{3} + 156\cdot 173^{4} +O\left(173^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 57 + 15\cdot 173 + 94\cdot 173^{2} + 64\cdot 173^{3} + 143\cdot 173^{4} +O\left(173^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 73 + 28\cdot 173 + 88\cdot 173^{2} + 171\cdot 173^{3} + 2\cdot 173^{4} +O\left(173^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 169 + 53\cdot 173 + 72\cdot 173^{2} + 163\cdot 173^{3} + 42\cdot 173^{4} +O\left(173^{ 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.