Properties

Label 5.2e4_44171e2.10t13.1c1
Dimension 5
Group $S_5$
Conductor $ 2^{4} \cdot 44171^{2}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$5$
Group:$S_5$
Conductor:$31217235856= 2^{4} \cdot 44171^{2} $
Artin number field: Splitting field of $f= x^{5} - 6 x^{3} - x^{2} + 5 x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_5$
Parity: Even
Determinant: 1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 269 }$ to precision 5.
Roots: \[ \begin{aligned} r_{ 1 } &= 23 + 133\cdot 269 + 44\cdot 269^{2} + 33\cdot 269^{3} + 60\cdot 269^{4} +O\left(269^{ 5 }\right) \\ r_{ 2 } &= 136 + 243\cdot 269 + 187\cdot 269^{2} + 260\cdot 269^{3} + 125\cdot 269^{4} +O\left(269^{ 5 }\right) \\ r_{ 3 } &= 176 + 64\cdot 269 + 77\cdot 269^{2} + 41\cdot 269^{3} + 77\cdot 269^{4} +O\left(269^{ 5 }\right) \\ r_{ 4 } &= 224 + 37\cdot 269 + 111\cdot 269^{2} + 8\cdot 269^{3} + 49\cdot 269^{4} +O\left(269^{ 5 }\right) \\ r_{ 5 } &= 248 + 58\cdot 269 + 117\cdot 269^{2} + 194\cdot 269^{3} + 225\cdot 269^{4} +O\left(269^{ 5 }\right) \\ \end{aligned}\]

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$$1$$()$$5$
$10$$2$$(1,2)$$1$
$15$$2$$(1,2)(3,4)$$1$
$20$$3$$(1,2,3)$$-1$
$30$$4$$(1,2,3,4)$$-1$
$24$$5$$(1,2,3,4,5)$$0$
$20$$6$$(1,2,3)(4,5)$$1$
The blue line marks the conjugacy class containing complex conjugation.