Properties

Label 5.107_367.6t16.1c1
Dimension 5
Group $S_6$
Conductor $ 107 \cdot 367 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$5$
Group:$S_6$
Conductor:$39269= 107 \cdot 367 $
Artin number field: Splitting field of $f= x^{6} - x^{5} - 2 x^{4} + 2 x^{3} - 2 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_6$
Parity: Even
Determinant: 1.107_367.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 577 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 135 + 47\cdot 577 + 291\cdot 577^{2} + 559\cdot 577^{3} + 163\cdot 577^{4} +O\left(577^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 197 + 340\cdot 577 + 441\cdot 577^{2} + 384\cdot 577^{3} + 239\cdot 577^{4} +O\left(577^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 202 + 5\cdot 577 + 94\cdot 577^{2} + 575\cdot 577^{3} + 150\cdot 577^{4} +O\left(577^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 368 + 55\cdot 577 + 510\cdot 577^{2} + 404\cdot 577^{3} +O\left(577^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 382 + 576\cdot 577 + 564\cdot 577^{2} + 65\cdot 577^{3} + 262\cdot 577^{4} +O\left(577^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 448 + 128\cdot 577 + 406\cdot 577^{2} + 317\cdot 577^{3} + 336\cdot 577^{4} +O\left(577^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

Cycle notation
$(1,2)$
$(1,2,3,4,5,6)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$5$
$15$$2$$(1,2)(3,4)(5,6)$$-1$
$15$$2$$(1,2)$$3$
$45$$2$$(1,2)(3,4)$$1$
$40$$3$$(1,2,3)(4,5,6)$$-1$
$40$$3$$(1,2,3)$$2$
$90$$4$$(1,2,3,4)(5,6)$$-1$
$90$$4$$(1,2,3,4)$$1$
$144$$5$$(1,2,3,4,5)$$0$
$120$$6$$(1,2,3,4,5,6)$$-1$
$120$$6$$(1,2,3)(4,5)$$0$
The blue line marks the conjugacy class containing complex conjugation.