Properties

Label 4.5515586099981504.10t12.a.a
Dimension 4
Group $S_5$
Conductor $ 2^{6} \cdot 44171^{3}$
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$4$
Group:$S_5$
Conductor:$5515586099981504= 2^{6} \cdot 44171^{3} $
Artin number field: Splitting field of 5.5.176684.1 defined by $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: 1.176684.2t1.a.a
Projective image: $S_5$
Projective field: Galois closure of 5.5.176684.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 269 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 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)$

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.