Properties

Label 3.2e4_89.4t5.1c1
Dimension 3
Group $S_4$
Conductor $ 2^{4} \cdot 89 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$3$
Group:$S_4$
Conductor:$1424= 2^{4} \cdot 89 $
Artin number field: Splitting field of $f= x^{4} + x^{2} - 2 x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_4$
Parity: Odd
Determinant: 1.2e2_89.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 83 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 3 + 20\cdot 83 + 59\cdot 83^{2} + 64\cdot 83^{3} + 75\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 32 + 74\cdot 83 + 19\cdot 83^{2} + 23\cdot 83^{3} + 70\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 63 + 5\cdot 83 + 16\cdot 83^{2} + 15\cdot 83^{3} + 30\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 68 + 65\cdot 83 + 70\cdot 83^{2} + 62\cdot 83^{3} + 72\cdot 83^{4} +O\left(83^{ 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.