Properties

Label 3.761.4t5.1c1
Dimension 3
Group $S_4$
Conductor $ 761 $
Root number 1
Frobenius-Schur indicator 1

Related objects

Learn more about

Basic invariants

Dimension:$3$
Group:$S_4$
Conductor:$761 $
Artin number field: Splitting field of $f= x^{4} - x^{3} + x^{2} + 2 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_4$
Parity: Even
Determinant: 1.761.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 139 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 21 + 41\cdot 139 + 31\cdot 139^{2} + 41\cdot 139^{3} + 108\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 58 + 83\cdot 139 + 8\cdot 139^{2} + 3\cdot 139^{3} + 77\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 68 + 4\cdot 139 + 61\cdot 139^{2} + 111\cdot 139^{3} + 50\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 132 + 9\cdot 139 + 38\cdot 139^{2} + 122\cdot 139^{3} + 41\cdot 139^{4} +O\left(139^{ 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.