Properties

Label 3.751e2.6t8.1c1
Dimension 3
Group $S_4$
Conductor $ 751^{2}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$3$
Group:$S_4$
Conductor:$564001= 751^{2} $
Artin number field: Splitting field of $f= x^{4} - 2 x^{3} + x^{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 $\Q_{ 163 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 32 + 98\cdot 163 + 133\cdot 163^{2} + 94\cdot 163^{3} + 70\cdot 163^{4} +O\left(163^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 34 + 7\cdot 163 + 75\cdot 163^{2} + 5\cdot 163^{3} + 99\cdot 163^{4} +O\left(163^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 118 + 112\cdot 163 + 142\cdot 163^{2} + 115\cdot 163^{3} + 75\cdot 163^{4} +O\left(163^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 144 + 107\cdot 163 + 137\cdot 163^{2} + 109\cdot 163^{3} + 80\cdot 163^{4} +O\left(163^{ 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.