Properties

Label 4.3e4_73.5t5.1c1
Dimension 4
Group $S_5$
Conductor $ 3^{4} \cdot 73 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$S_5$
Conductor:$5913= 3^{4} \cdot 73 $
Artin number field: Splitting field of $f= x^{5} - x^{4} + 2 x^{3} - x^{2} + x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_5$
Parity: Even
Determinant: 1.73.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 457 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 45 + 171\cdot 457 + 240\cdot 457^{2} + 61\cdot 457^{3} + 205\cdot 457^{4} +O\left(457^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 102 + 353\cdot 457 + 118\cdot 457^{2} + 178\cdot 457^{3} + 93\cdot 457^{4} +O\left(457^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 149 + 343\cdot 457 + 127\cdot 457^{2} + 182\cdot 457^{3} + 368\cdot 457^{4} +O\left(457^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 237 + 251\cdot 457 + 79\cdot 457^{2} + 429\cdot 457^{3} + 235\cdot 457^{4} +O\left(457^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 382 + 251\cdot 457 + 347\cdot 457^{2} + 62\cdot 457^{3} + 11\cdot 457^{4} +O\left(457^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 5 }$

Cycle notation
$(1,2)$
$(1,2,3,4,5)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 5 }$ Character value
$1$$1$$()$$4$
$10$$2$$(1,2)$$2$
$15$$2$$(1,2)(3,4)$$0$
$20$$3$$(1,2,3)$$1$
$30$$4$$(1,2,3,4)$$0$
$24$$5$$(1,2,3,4,5)$$-1$
$20$$6$$(1,2,3)(4,5)$$-1$
The blue line marks the conjugacy class containing complex conjugation.