Properties

Label 5.236549e3.6t14.1c1
Dimension 5
Group $S_5$
Conductor $ 236549^{3}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$5$
Group:$S_5$
Conductor:$13236200869377149= 236549^{3} $
Artin number field: Splitting field of $f= x^{5} - x^{4} - 6 x^{3} + 7 x^{2} + 2 x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $\PGL(2,5)$
Parity: Even
Determinant: 1.236549.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 461 }$ to precision 5.
Roots: \[ \begin{aligned} r_{ 1 } &= 98 + 99\cdot 461 + 277\cdot 461^{2} + 443\cdot 461^{3} + 317\cdot 461^{4} +O\left(461^{ 5 }\right) \\ r_{ 2 } &= 203 + 279\cdot 461 + 175\cdot 461^{2} + 315\cdot 461^{3} + 430\cdot 461^{4} +O\left(461^{ 5 }\right) \\ r_{ 3 } &= 247 + 346\cdot 461 + 218\cdot 461^{2} + 382\cdot 461^{3} + 138\cdot 461^{4} +O\left(461^{ 5 }\right) \\ r_{ 4 } &= 386 + 441\cdot 461 + 9\cdot 461^{2} + 144\cdot 461^{3} + 38\cdot 461^{4} +O\left(461^{ 5 }\right) \\ r_{ 5 } &= 450 + 215\cdot 461 + 240\cdot 461^{2} + 97\cdot 461^{3} + 457\cdot 461^{4} +O\left(461^{ 5 }\right) \\ \end{aligned}\]

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.