Properties

Label 3.3_2351.4t5.1c1
Dimension 3
Group $S_4$
Conductor $ 3 \cdot 2351 $
Root number 1
Frobenius-Schur indicator 1

Related objects

Learn more about

Basic invariants

Dimension:$3$
Group:$S_4$
Conductor:$7053= 3 \cdot 2351 $
Artin number field: Splitting field of $f= x^{4} - 2 x^{3} - 4 x^{2} + 3 x + 3 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_4$
Parity: Even
Determinant: 1.3_2351.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 229 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 9 + 190\cdot 229 + 70\cdot 229^{2} + 68\cdot 229^{3} + 161\cdot 229^{4} +O\left(229^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 17 + 132\cdot 229 + 214\cdot 229^{2} + 200\cdot 229^{3} + 51\cdot 229^{4} +O\left(229^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 77 + 222\cdot 229 + 157\cdot 229^{2} + 103\cdot 229^{3} + 160\cdot 229^{4} +O\left(229^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 128 + 142\cdot 229 + 14\cdot 229^{2} + 85\cdot 229^{3} + 84\cdot 229^{4} +O\left(229^{ 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.