Properties

Label 4.2e2_1291.5t5.1c1
Dimension 4
Group $S_5$
Conductor $ 2^{2} \cdot 1291 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$S_5$
Conductor:$5164= 2^{2} \cdot 1291 $
Artin number field: Splitting field of $f= x^{5} - x^{4} - x^{3} + 2 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_5$
Parity: Even
Determinant: 1.2e2_1291.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 283 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 57 + 67\cdot 283 + 157\cdot 283^{2} + 9\cdot 283^{3} + 61\cdot 283^{4} +O\left(283^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 154 + 11\cdot 283 + 124\cdot 283^{2} + 196\cdot 283^{3} + 198\cdot 283^{4} +O\left(283^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 180 + 118\cdot 283 + 256\cdot 283^{2} + 131\cdot 283^{3} + 99\cdot 283^{4} +O\left(283^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 198 + 254\cdot 283 + 20\cdot 283^{2} + 86\cdot 283^{3} + 137\cdot 283^{4} +O\left(283^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 261 + 113\cdot 283 + 7\cdot 283^{2} + 142\cdot 283^{3} + 69\cdot 283^{4} +O\left(283^{ 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.